Theory AOT_PLM

1(*<*)
2theory AOT_PLM
3  imports AOT_axioms
4begin
5(*>*)
6
7section‹The Deductive System PLM›
8
9(* constrain sledgehammer to the abstraction layer *)
10unbundle AOT_no_atp
11
12(* To enable meta syntax: *)
13(*interpretation AOT_meta_syntax.*)
14(* To disable meta syntax: *)
15interpretation AOT_no_meta_syntax.
16
17(* To enable AOT syntax (takes precedence over meta syntax; can be done locally using "including" or "include"): *)
18unbundle AOT_syntax
19(* To disable AOT syntax (restoring meta syntax or no syntax; can be done locally using "including" or "include"): *)
20(* unbundle AOT_no_syntax *)
21
22AOT_theorem "modus-ponens": assumes ‹φ› and ‹φ → ψ› shows ‹ψ›
23  using assms by (simp add: AOT_sem_imp) (* NOTE: semantics needed *)
24lemmas MP = "modus-ponens"
25
26AOT_theorem "non-con-thm-thm": assumes ‹⊢□ φ› shows ‹⊢ φ›
27  using assms by simp
28
29AOT_theorem "vdash-properties:1[1]": assumes ‹φ ∈ Λ› shows ‹⊢ φ›
30  using assms unfolding AOT_model_act_axiom_def by blast (* NOTE: semantics needed *)
31
32attribute_setup act_axiom_inst =
33  ‹Scan.succeed (Thm.rule_attribute [] (K (fn thm => thm RS @{thm "vdash-properties:1[1]"})))›
34  "Instantiate modally fragile axiom as modally fragile theorem."
35
36AOT_theorem "vdash-properties:1[2]": assumes ‹φ ∈ Λ□› shows ‹⊢□ φ›
37  using assms unfolding AOT_model_axiom_def by blast (* NOTE: semantics needed *)
38
39attribute_setup axiom_inst =
40  ‹Scan.succeed (Thm.rule_attribute [] (K (fn thm => thm RS @{thm "vdash-properties:1[2]"})))›
41  "Instantiate axiom as theorem."
42
43method cqt_2_lambda_inst_prover = (fast intro: AOT_instance_of_cqt_2_intro)
44method "cqt:2[lambda]" = (rule "cqt:2[lambda]"[axiom_inst]; cqt_2_lambda_inst_prover)
45
46AOT_theorem "vdash-properties:3": assumes ‹⊢□ φ› shows ‹Γ ⊢ φ›
47  using assms by blast
48
49AOT_theorem "vdash-properties:5": assumes ‹Γ1 ⊢ φ› and ‹Γ2 ⊢ φ → ψ› shows ‹Γ1, Γ2 ⊢ ψ›
50  using MP assms by blast
51
52AOT_theorem "vdash-properties:6": assumes ‹φ› and ‹φ → ψ› shows ‹ψ›
53  using MP assms by blast
54
55AOT_theorem "vdash-properties:8": assumes ‹Γ ⊢ φ› and ‹φ ⊢ ψ› shows ‹Γ ⊢ ψ›
56  using assms by argo
57
58AOT_theorem "vdash-properties:9": assumes ‹φ› shows ‹ψ → φ›
59  using MP "pl:1"[axiom_inst] assms by blast
60
61AOT_theorem "vdash-properties:10": assumes ‹φ → ψ› and ‹φ› shows ‹ψ›
62  using MP assms by blast
63lemmas "→E" = "vdash-properties:10"
64
65AOT_theorem "rule-gen": assumes ‹for arbitrary α: φ{α}› shows ‹∀α φ{α}›
66  using assms by (metis AOT_var_of_term_inverse AOT_sem_denotes AOT_sem_forall) (* NOTE: semantics needed *)
67lemmas GEN = "rule-gen"
68
69AOT_theorem "RN[prem]": assumes ‹Γ ⊢□ φ› shows ‹□Γ ⊢□ □φ›
70  by (meson AOT_sem_box assms image_iff) (* NOTE: semantics needed *)
71AOT_theorem RN: assumes ‹⊢□ φ› shows ‹□φ›
72  using "RN[prem]" assms by blast
73
74AOT_axiom "df-rules-formulas[1]": assumes ‹φ ≡df ψ› shows ‹φ → ψ›
75  using assms by (simp_all add: AOT_model_axiomI AOT_model_equiv_def AOT_sem_imp) (* NOTE: semantics needed *)
76AOT_axiom "df-rules-formulas[2]": assumes ‹φ ≡df ψ› shows ‹ψ → φ›
77  using assms by (simp_all add: AOT_model_axiomI AOT_model_equiv_def AOT_sem_imp) (* NOTE: semantics needed *)
78(* NOTE: for convenience also state the above as regular theorems *)
79AOT_theorem "df-rules-formulas[3]": assumes ‹φ ≡df ψ› shows ‹φ → ψ›
80  using "df-rules-formulas[1]"[axiom_inst, OF assms].
81AOT_theorem "df-rules-formulas[4]": assumes ‹φ ≡df ψ› shows ‹ψ → φ›
82  using "df-rules-formulas[2]"[axiom_inst, OF assms].
83
84
85AOT_axiom "df-rules-terms[1]":
86  assumes ‹τ{α1...αn} =df σ{α1...αn}›
87  shows ‹(σ{τ1...τn}↓ → τ{τ1...τn} = σ{τ1...τn}) & (¬σ{τ1...τn}↓ → ¬τ{τ1...τn}↓)›
88  using assms by (simp add: AOT_model_axiomI AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def) (* NOTE: semantics needed *)
89AOT_axiom "df-rules-terms[2]":
90  assumes ‹τ =df σ›
91  shows ‹(σ↓ → τ = σ) & (¬σ↓ → ¬τ↓)›
92  by (metis "df-rules-terms[1]" case_unit_Unity assms)
93(* NOTE: for convenience also state the above as regular theorems *)
94AOT_theorem "df-rules-terms[3]":
95  assumes ‹τ{α1...αn} =df σ{α1...αn}›
96  shows ‹(σ{τ1...τn}↓ → τ{τ1...τn} = σ{τ1...τn}) & (¬σ{τ1...τn}↓ → ¬τ{τ1...τn}↓)›
97  using "df-rules-terms[1]"[axiom_inst, OF assms].
98AOT_theorem "df-rules-terms[4]":
99  assumes ‹τ =df σ›
100  shows ‹(σ↓ → τ = σ) & (¬σ↓ → ¬τ↓)›
101  using "df-rules-terms[2]"[axiom_inst, OF assms].
102
103
104AOT_theorem "if-p-then-p": ‹φ → φ›
105  by (meson "pl:1"[axiom_inst] "pl:2"[axiom_inst] MP)
106
107AOT_theorem "deduction-theorem": assumes ‹φ ⊢ ψ› shows ‹φ → ψ›
108  using assms by (simp add: AOT_sem_imp) (* NOTE: semantics needed *)
109lemmas CP = "deduction-theorem"
110lemmas "→I" = "deduction-theorem"
111
112AOT_theorem "ded-thm-cor:1": assumes ‹Γ1 ⊢ φ → ψ› and ‹Γ2 ⊢ ψ → χ› shows ‹Γ1, Γ2 ⊢ φ → χ›
113  using "→E" "→I" assms by blast
114AOT_theorem "ded-thm-cor:2": assumes ‹Γ1 ⊢ φ → (ψ → χ)› and ‹Γ2 ⊢ ψ› shows ‹Γ1, Γ2 ⊢ φ → χ›
115  using "→E" "→I" assms by blast
116
117AOT_theorem "ded-thm-cor:3": assumes ‹φ → ψ› and ‹ψ → χ› shows ‹φ → χ›
118  using "→E" "→I" assms by blast
119declare "ded-thm-cor:3"[trans]
120AOT_theorem "ded-thm-cor:4": assumes ‹φ → (ψ → χ)› and ‹ψ› shows ‹φ → χ›
121  using "→E" "→I" assms by blast
122
123lemmas "Hypothetical Syllogism" = "ded-thm-cor:3"
124
125AOT_theorem "useful-tautologies:1": ‹¬¬φ → φ›
126  by (metis "pl:3"[axiom_inst] "→I" "Hypothetical Syllogism")
127AOT_theorem "useful-tautologies:2": ‹φ → ¬¬φ›
128  by (metis "pl:3"[axiom_inst] "→I" "ded-thm-cor:4")
129AOT_theorem "useful-tautologies:3": ‹¬φ → (φ → ψ)›
130  by (meson "ded-thm-cor:4" "pl:3"[axiom_inst] "→I")
131AOT_theorem "useful-tautologies:4": ‹(¬ψ → ¬φ) → (φ → ψ)›
132  by (meson "pl:3"[axiom_inst] "Hypothetical Syllogism" "→I")
133AOT_theorem "useful-tautologies:5": ‹(φ → ψ) → (¬ψ → ¬φ)›
134  by (metis "useful-tautologies:4" "Hypothetical Syllogism" "→I")
135
136AOT_theorem "useful-tautologies:6": ‹(φ → ¬ψ) → (ψ → ¬φ)›
137  by (metis "→I" MP "useful-tautologies:4")
138
139AOT_theorem "useful-tautologies:7": ‹(¬φ → ψ) → (¬ψ → φ)›
140  by (metis "→I" MP "useful-tautologies:3" "useful-tautologies:5")
141
142AOT_theorem "useful-tautologies:8": ‹φ → (¬ψ → ¬(φ → ψ))›
143  by (metis "→I" MP "useful-tautologies:5")
144
145AOT_theorem "useful-tautologies:9": ‹(φ → ψ) → ((¬φ → ψ) → ψ)›
146  by (metis "→I" MP "useful-tautologies:6")
147
148AOT_theorem "useful-tautologies:10": ‹(φ → ¬ψ) → ((φ → ψ) → ¬φ)›
149  by (metis "→I" MP "pl:3"[axiom_inst])
150
151AOT_theorem "dn-i-e:1": assumes ‹φ› shows ‹¬¬φ›
152  using MP "useful-tautologies:2" assms by blast
153lemmas "¬¬I" = "dn-i-e:1"
154AOT_theorem "dn-i-e:2": assumes ‹¬¬φ› shows ‹φ›
155  using MP "useful-tautologies:1" assms by blast
156lemmas "¬¬E" = "dn-i-e:2"
157
158AOT_theorem "modus-tollens:1": assumes ‹φ → ψ› and ‹¬ψ› shows ‹¬φ›
159  using MP "useful-tautologies:5" assms by blast
160AOT_theorem "modus-tollens:2": assumes ‹φ → ¬ψ› and ‹ψ› shows ‹¬φ›
161  using "¬¬I" "modus-tollens:1" assms by blast
162lemmas MT = "modus-tollens:1" "modus-tollens:2"
163
164AOT_theorem "contraposition:1[1]": assumes ‹φ → ψ› shows ‹¬ψ → ¬φ›
165  using "→I" MT(1) assms by blast
166AOT_theorem "contraposition:1[2]": assumes ‹¬ψ → ¬φ› shows ‹φ → ψ›
167  using "→I" "¬¬E" MT(2) assms by blast
168
169AOT_theorem "contraposition:2": assumes ‹φ → ¬ψ› shows ‹ψ → ¬φ›
170  using "→I" MT(2) assms by blast
171
172(* TODO: this is actually a mixture of the two variants given in PLM; adjust. *)
173AOT_theorem "reductio-aa:1":
174  assumes ‹¬φ ⊢ ¬ψ› and ‹¬φ ⊢ ψ› shows ‹φ›
175  using "→I" "¬¬E" MT(2) assms by blast
176AOT_theorem "reductio-aa:2":
177  assumes ‹φ ⊢ ¬ψ› and ‹φ ⊢ ψ› shows ‹¬φ›
178  using "reductio-aa:1" assms by blast
179lemmas "RAA" = "reductio-aa:1" "reductio-aa:2"
180
181AOT_theorem "exc-mid": ‹φ ∨ ¬φ›
182  using "df-rules-formulas[4]" "if-p-then-p" MP "conventions:2" by blast
183
184AOT_theorem "non-contradiction": ‹¬(φ & ¬φ)›
185  using "df-rules-formulas[3]" MT(2) "useful-tautologies:2" "conventions:1" by blast
186
187AOT_theorem "con-dis-taut:1": ‹(φ & ψ) → φ›
188  by (meson "→I" "df-rules-formulas[3]" MP RAA(1) "conventions:1")
189AOT_theorem "con-dis-taut:2": ‹(φ & ψ) → ψ›
190  by (metis "→I" "df-rules-formulas[3]" MT(2) RAA(2) "¬¬E" "conventions:1")
191lemmas "Conjunction Simplification" = "con-dis-taut:1" "con-dis-taut:2"
192
193AOT_theorem "con-dis-taut:3": ‹φ → (φ ∨ ψ)›
194  by (meson "contraposition:1[2]" "df-rules-formulas[4]" MP "→I" "conventions:2")
195AOT_theorem "con-dis-taut:4": ‹ψ → (φ ∨ ψ)›
196  using "Hypothetical Syllogism" "df-rules-formulas[4]" "pl:1"[axiom_inst] "conventions:2" by blast
197lemmas "Disjunction Addition" = "con-dis-taut:3" "con-dis-taut:4"
198
199AOT_theorem "con-dis-taut:5": ‹φ → (ψ → (φ & ψ))›
200  by (metis "contraposition:2" "Hypothetical Syllogism" "→I" "df-rules-formulas[4]" "conventions:1")
201lemmas Adjunction = "con-dis-taut:5"
202
203AOT_theorem "con-dis-taut:6": ‹(φ & φ) ≡ φ›
204  by (metis Adjunction "→I" "df-rules-formulas[4]" MP "Conjunction Simplification"(1) "conventions:3")
205lemmas "Idempotence of &" = "con-dis-taut:6"
206
207AOT_theorem "con-dis-taut:7": ‹(φ ∨ φ) ≡ φ›
208proof -
209  {
210    AOT_assume ‹φ ∨ φ›
211    AOT_hence ‹¬φ → φ›
212      using "conventions:2"[THEN "df-rules-formulas[3]"] MP by blast
213    AOT_hence ‹φ› using "if-p-then-p" RAA(1) MP by blast
214  }
215  moreover {
216    AOT_assume ‹φ›
217    AOT_hence ‹φ ∨ φ› using "Disjunction Addition"(1) MP by blast
218  }
219  ultimately AOT_show ‹(φ ∨ φ) ≡ φ›
220    using "conventions:3"[THEN "df-rules-formulas[4]"] MP
221    by (metis Adjunction "→I")
222qed
223lemmas "Idempotence of ∨" = "con-dis-taut:7"
224
225
226AOT_theorem "con-dis-i-e:1": assumes ‹φ› and ‹ψ› shows ‹φ & ψ›
227  using Adjunction MP assms by blast
228lemmas "&I" = "con-dis-i-e:1"
229
230AOT_theorem "con-dis-i-e:2:a": assumes ‹φ & ψ› shows ‹φ›
231  using "Conjunction Simplification"(1) MP assms by blast
232AOT_theorem "con-dis-i-e:2:b": assumes ‹φ & ψ› shows ‹ψ›
233  using "Conjunction Simplification"(2) MP assms by blast
234lemmas "&E" = "con-dis-i-e:2:a" "con-dis-i-e:2:b"
235
236AOT_theorem "con-dis-i-e:3:a": assumes ‹φ› shows ‹φ ∨ ψ›
237  using "Disjunction Addition"(1) MP assms by blast
238AOT_theorem "con-dis-i-e:3:b": assumes ‹ψ› shows ‹φ ∨ ψ›
239  using "Disjunction Addition"(2) MP assms by blast
240AOT_theorem "con-dis-i-e:3:c": assumes ‹φ ∨ ψ› and ‹φ → χ› and ‹ψ → Θ› shows ‹χ ∨ Θ›
241  by (metis "con-dis-i-e:3:a" "Disjunction Addition"(2) "df-rules-formulas[3]" MT(1) RAA(1) "conventions:2" assms)
242lemmas "∨I" = "con-dis-i-e:3:a" "con-dis-i-e:3:b" "con-dis-i-e:3:c"
243
244AOT_theorem "con-dis-i-e:4:a": assumes ‹φ ∨ ψ› and ‹φ → χ› and ‹ψ → χ› shows ‹χ›
245  by (metis MP RAA(2) "df-rules-formulas[3]" "conventions:2" assms)
246AOT_theorem "con-dis-i-e:4:b": assumes ‹φ ∨ ψ› and ‹¬φ› shows ‹ψ›
247  using "con-dis-i-e:4:a" RAA(1) "→I" assms by blast
248AOT_theorem "con-dis-i-e:4:c": assumes ‹φ ∨ ψ› and ‹¬ψ› shows ‹φ›
249  using "con-dis-i-e:4:a" RAA(1) "→I" assms by blast
250lemmas "∨E" = "con-dis-i-e:4:a" "con-dis-i-e:4:b" "con-dis-i-e:4:c"
251
252AOT_theorem "raa-cor:1": assumes ‹¬φ ⊢ ψ & ¬ψ› shows ‹φ›
253  using "&E" "∨E"(3) "∨I"(2) RAA(2) assms by blast
254AOT_theorem "raa-cor:2": assumes ‹φ ⊢ ψ & ¬ψ› shows ‹¬φ›
255  using "raa-cor:1" assms by blast
256AOT_theorem "raa-cor:3": assumes ‹φ› and ‹¬ψ ⊢ ¬φ› shows ‹ψ›
257  using RAA assms by blast
258AOT_theorem "raa-cor:4": assumes ‹¬φ› and ‹¬ψ ⊢ φ› shows ‹ψ›
259  using RAA assms by blast
260AOT_theorem "raa-cor:5": assumes ‹φ› and ‹ψ ⊢ ¬φ› shows ‹¬ψ›
261  using RAA assms by blast
262AOT_theorem "raa-cor:6": assumes ‹¬φ› and ‹ψ ⊢ φ› shows ‹¬ψ›
263  using RAA assms by blast
264
265(* TODO: note these need manual introduction rules *)
266AOT_theorem "oth-class-taut:1:a": ‹(φ → ψ) ≡ ¬(φ & ¬ψ)›
267  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
268     (metis "&E" "&I" "raa-cor:3" "→I" MP)
269AOT_theorem "oth-class-taut:1:b": ‹¬(φ → ψ) ≡ (φ & ¬ψ)›
270  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
271     (metis "&E" "&I" "raa-cor:3" "→I" MP)
272AOT_theorem "oth-class-taut:1:c": ‹(φ → ψ) ≡ (¬φ ∨ ψ)›
273  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
274     (metis "&I" "∨I"(1, 2) "∨E"(3) "→I" MP "raa-cor:1")
275
276AOT_theorem "oth-class-taut:2:a": ‹(φ & ψ) ≡ (ψ & φ)›
277  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
278     (meson "&I" "&E" "→I")
279lemmas "Commutativity of &" = "oth-class-taut:2:a"
280AOT_theorem "oth-class-taut:2:b": ‹(φ & (ψ & χ)) ≡ ((φ & ψ) & χ)›
281  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
282     (metis "&I" "&E" "→I")
283lemmas "Associativity of &" = "oth-class-taut:2:b"
284AOT_theorem "oth-class-taut:2:c": ‹(φ ∨ ψ) ≡ (ψ ∨ φ)›
285  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
286     (metis "&I" "∨I"(1, 2) "∨E"(1) "→I")
287lemmas "Commutativity of ∨" = "oth-class-taut:2:c"
288AOT_theorem "oth-class-taut:2:d": ‹(φ ∨ (ψ ∨ χ)) ≡ ((φ ∨ ψ) ∨ χ)›
289  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
290     (metis "&I" "∨I"(1, 2) "∨E"(1) "→I")
291lemmas "Associativity of ∨" = "oth-class-taut:2:d"
292AOT_theorem "oth-class-taut:2:e": ‹(φ ≡ ψ) ≡ (ψ ≡ φ)›
293  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]; rule "&I";
294      metis "&I" "df-rules-formulas[4]" "conventions:3" "&E" "Hypothetical Syllogism" "→I" "df-rules-formulas[3]")
295lemmas "Commutativity of ≡" = "oth-class-taut:2:e"
296AOT_theorem "oth-class-taut:2:f": ‹(φ ≡ (ψ ≡ χ)) ≡ ((φ ≡ ψ) ≡ χ)›
297  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
298        "→I" "→E" "&E" "&I"
299  by metis
300lemmas "Associativity of ≡" = "oth-class-taut:2:f"
301
302AOT_theorem "oth-class-taut:3:a": ‹φ ≡ φ›
303  using "&I" "vdash-properties:6" "if-p-then-p" "df-rules-formulas[4]" "conventions:3" by blast
304AOT_theorem "oth-class-taut:3:b": ‹φ ≡ ¬¬φ›
305  using "&I" "useful-tautologies:1" "useful-tautologies:2" "vdash-properties:6" "df-rules-formulas[4]" "conventions:3" by blast
306AOT_theorem "oth-class-taut:3:c": ‹¬(φ ≡ ¬φ)›
307  by (metis "&E" "→E" RAA "df-rules-formulas[3]" "conventions:3")
308
309AOT_theorem "oth-class-taut:4:a": ‹(φ → ψ) → ((ψ → χ) → (φ → χ))›
310  by (metis "→E" "→I")
311AOT_theorem "oth-class-taut:4:b": ‹(φ ≡ ψ) ≡ (¬φ ≡ ¬ψ)›
312  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
313        "→I" "→E" "&E" "&I" RAA by metis
314AOT_theorem "oth-class-taut:4:c": ‹(φ ≡ ψ) → ((φ → χ) ≡ (ψ → χ))›
315  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
316        "→I" "→E" "&E" "&I" by metis
317AOT_theorem "oth-class-taut:4:d": ‹(φ ≡ ψ) → ((χ → φ) ≡ (χ → ψ))›
318  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
319        "→I" "→E" "&E" "&I" by metis
320AOT_theorem "oth-class-taut:4:e": ‹(φ ≡ ψ) → ((φ & χ) ≡ (ψ & χ))›
321  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
322        "→I" "→E" "&E" "&I" by metis
323AOT_theorem "oth-class-taut:4:f": ‹(φ ≡ ψ) → ((χ & φ) ≡ (χ & ψ))›
324  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
325        "→I" "→E" "&E" "&I" by metis
326(* TODO: nicer proof *)
327AOT_theorem "oth-class-taut:4:g": ‹(φ ≡ ψ) ≡ ((φ & ψ) ∨ (¬φ & ¬ψ))›
328  apply (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]; rule "&I"; rule "→I")
329   apply (drule "conventions:3"[THEN "df-rules-formulas[3]", THEN "→E"])
330   apply (metis "&I" "&E" "∨I"(1,2) MT(1) "raa-cor:3")
331  apply (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]; rule "&I"; rule "→I")
332  using "&E" "∨E"(2) "raa-cor:3" by blast+
333AOT_theorem "oth-class-taut:4:h": ‹¬(φ ≡ ψ) ≡ ((φ & ¬ψ) ∨ (¬φ & ψ))›
334  apply (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]; rule "&I"; rule "→I")
335  apply (metis "&I" "∨I"(1, 2) "→I" MT(1) "df-rules-formulas[4]" "raa-cor:3" "conventions:3")
336  by (metis "&E" "∨E"(2) "→E" "df-rules-formulas[3]" "raa-cor:3" "conventions:3")
337AOT_theorem "oth-class-taut:5:a": ‹(φ & ψ) ≡ ¬(¬φ ∨ ¬ψ)›
338  using "conventions:3"[THEN "df-rules-formulas[4]"]
339        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
340AOT_theorem "oth-class-taut:5:b": ‹(φ ∨ ψ) ≡ ¬(¬φ & ¬ψ)›
341  using "conventions:3"[THEN "df-rules-formulas[4]"]
342        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
343AOT_theorem "oth-class-taut:5:c": ‹¬(φ & ψ) ≡ (¬φ ∨ ¬ψ)›
344  using "conventions:3"[THEN "df-rules-formulas[4]"]
345        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
346AOT_theorem "oth-class-taut:5:d": ‹¬(φ ∨ ψ) ≡ (¬φ & ¬ψ)›
347  using "conventions:3"[THEN "df-rules-formulas[4]"]
348        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
349
350lemmas DeMorgan = "oth-class-taut:5:c" "oth-class-taut:5:d"
351
352AOT_theorem "oth-class-taut:6:a": ‹(φ & (ψ ∨ χ)) ≡ ((φ & ψ) ∨ (φ & χ))›
353  using "conventions:3"[THEN "df-rules-formulas[4]"]
354        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
355AOT_theorem "oth-class-taut:6:b": ‹(φ ∨ (ψ & χ)) ≡ ((φ ∨ ψ) & (φ ∨ χ))›
356  using "conventions:3"[THEN "df-rules-formulas[4]"]
357        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
358
359AOT_theorem "oth-class-taut:7:a": ‹((φ & ψ) → χ) → (φ → (ψ → χ))›
360  by (metis "&I" "→E" "→I")
361lemmas Exportation = "oth-class-taut:7:a"
362AOT_theorem "oth-class-taut:7:b": ‹(φ → (ψ →χ)) → ((φ & ψ) → χ)›
363  by (metis "&E" "→E" "→I")
364lemmas Importation = "oth-class-taut:7:b"
365
366AOT_theorem "oth-class-taut:8:a": ‹(φ → (ψ → χ)) ≡ (ψ → (φ → χ))›
367  using "conventions:3"[THEN "df-rules-formulas[4]"] "→I" "→E" "&E" "&I" by metis
368lemmas Permutation = "oth-class-taut:8:a"
369AOT_theorem "oth-class-taut:8:b": ‹(φ → ψ) → ((φ → χ) → (φ → (ψ & χ)))›
370  by (metis "&I" "→E" "→I")
371lemmas Composition = "oth-class-taut:8:b"
372AOT_theorem "oth-class-taut:8:c": ‹(φ → χ) → ((ψ → χ) → ((φ ∨ ψ) → χ))›
373  by (metis "∨E"(2) "→E" "→I" RAA(1))
374AOT_theorem "oth-class-taut:8:d": ‹((φ → ψ) & (χ → Θ)) → ((φ & χ) → (ψ & Θ))›
375  by (metis "&E" "&I" "→E" "→I")
376lemmas "Double Composition" = "oth-class-taut:8:d"
377AOT_theorem "oth-class-taut:8:e": ‹((φ & ψ) ≡ (φ & χ)) ≡ (φ → (ψ ≡ χ))›
378  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
379            "→I" "→E" "&E" "&I")
380AOT_theorem "oth-class-taut:8:f": ‹((φ & ψ) ≡ (χ & ψ)) ≡ (ψ → (φ ≡ χ))›
381  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
382            "→I" "→E" "&E" "&I")
383AOT_theorem "oth-class-taut:8:g": ‹(ψ ≡ χ) → ((φ ∨ ψ) ≡ (φ ∨ χ))›
384  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
385            "→I" "→E" "&E" "&I" "∨I" "∨E"(1))
386AOT_theorem "oth-class-taut:8:h": ‹(ψ ≡ χ) → ((ψ ∨ φ) ≡ (χ ∨ φ))›
387  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
388            "→I" "→E" "&E" "&I" "∨I" "∨E"(1))
389AOT_theorem "oth-class-taut:8:i": ‹(φ ≡ (ψ & χ)) → (ψ → (φ ≡ χ))›
390  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
391            "→I" "→E" "&E" "&I")
392
393AOT_theorem "intro-elim:1": assumes ‹φ ∨ ψ› and ‹φ ≡ χ› and ‹ψ ≡ Θ› shows ‹χ ∨ Θ›
394  by (metis assms "∨I"(1, 2) "∨E"(1) "conventions:3"[THEN "df-rules-formulas[3]"] "→I" "→E" "&E"(1))
395
396AOT_theorem "intro-elim:2": assumes ‹φ → ψ› and ‹ψ → φ› shows ‹φ ≡ ψ›
397  by (meson "&I" "conventions:3" "df-rules-formulas[4]" MP assms)
398lemmas "≡I" = "intro-elim:2"
399
400AOT_theorem "intro-elim:3:a": assumes ‹φ ≡ ψ› and ‹φ› shows ‹ψ›
401  by (metis "∨I"(1) "→I" "∨E"(1) "intro-elim:1" assms)
402AOT_theorem "intro-elim:3:b": assumes ‹φ ≡ ψ› and ‹ψ› shows ‹φ›
403  using "intro-elim:3:a" "Commutativity of ≡" assms by blast
404AOT_theorem "intro-elim:3:c": assumes ‹φ ≡ ψ› and ‹¬φ› shows ‹¬ψ›
405  using "intro-elim:3:b" "raa-cor:3" assms by blast
406AOT_theorem "intro-elim:3:d": assumes ‹φ ≡ ψ› and ‹¬ψ› shows ‹¬φ›
407  using "intro-elim:3:a" "raa-cor:3" assms by blast
408AOT_theorem "intro-elim:3:e": assumes ‹φ ≡ ψ› and ‹ψ ≡ χ› shows ‹φ ≡ χ›
409  by (metis "≡I" "→I" "intro-elim:3:a" "intro-elim:3:b" assms)
410declare "intro-elim:3:e"[trans]
411AOT_theorem "intro-elim:3:f": assumes ‹φ ≡ ψ› and ‹φ ≡ χ› shows ‹χ ≡ ψ›
412  by (metis "≡I" "→I" "intro-elim:3:a" "intro-elim:3:b" assms)
413lemmas "≡E" = "intro-elim:3:a" "intro-elim:3:b" "intro-elim:3:c" "intro-elim:3:d" "intro-elim:3:e" "intro-elim:3:f"
414
415declare "Commutativity of ≡"[THEN "≡E"(1), sym]
416
417AOT_theorem "rule-eq-df:1": assumes ‹φ ≡df ψ› shows ‹φ ≡ ψ›
418  by (simp add: "≡I" "df-rules-formulas[3]" "df-rules-formulas[4]" assms)
419lemmas "≡Df" = "rule-eq-df:1"
420AOT_theorem "rule-eq-df:2": assumes ‹φ ≡df ψ› and ‹φ› shows ‹ψ›
421  using "≡Df" "≡E"(1) assms by blast
422lemmas "≡dfE" = "rule-eq-df:2"
423AOT_theorem "rule-eq-df:3": assumes ‹φ ≡df ψ› and ‹ψ› shows ‹φ›
424  using "≡Df" "≡E"(2) assms by blast
425lemmas "≡dfI" = "rule-eq-df:3"
426
427AOT_theorem  "df-simplify:1": assumes ‹φ ≡ (ψ & χ)› and ‹ψ› shows ‹φ ≡ χ›
428  by (metis "&E"(2) "&I" "≡E"(1, 2) "≡I" "→I" assms)
429(* TODO: this is a slight variation from PLM *)
430AOT_theorem  "df-simplify:2": assumes ‹φ ≡ (ψ & χ)› and ‹χ› shows ‹φ ≡ ψ›
431  by (metis "&E"(1) "&I" "≡E"(1, 2) "≡I" "→I" assms)
432lemmas "≡S" = "df-simplify:1"  "df-simplify:2"
433
434AOT_theorem "rule-ui:1": assumes ‹∀α φ{α}› and ‹τ↓› shows ‹φ{τ}›
435  using "→E" "cqt:1"[axiom_inst] assms by blast
436AOT_theorem "rule-ui:2[const_var]": assumes ‹∀α φ{α}› shows ‹φ{β}›
437  by (simp add: "rule-ui:1" "cqt:2[const_var]"[axiom_inst] assms)
438(* TODO: precise proviso in PLM *)
439AOT_theorem "rule-ui:2[lambda]":
440  assumes ‹∀F φ{F}› and ‹INSTANCE_OF_CQT_2(ψ)›
441  shows ‹φ{[λν1...νn ψ{ν1...νn}]}›
442  by (simp add: "rule-ui:1" "cqt:2[lambda]"[axiom_inst] assms)
443AOT_theorem "rule-ui:3": assumes ‹∀α φ{α}› shows ‹φ{α}›
444  by (simp add: "rule-ui:2[const_var]" assms)
445lemmas "∀E" = "rule-ui:1" "rule-ui:2[const_var]" "rule-ui:2[lambda]" "rule-ui:3"
446
447AOT_theorem "cqt-orig:1[const_var]": ‹∀α φ{α} → φ{β}› by (simp add: "∀E"(2) "→I")
448AOT_theorem "cqt-orig:1[lambda]":
449  assumes ‹INSTANCE_OF_CQT_2(ψ)›
450  shows ‹∀F φ{F} → φ{[λν1...νn ψ{ν1...νn}]}›
451  by (simp add: "∀E"(3) "→I" assms)
452AOT_theorem "cqt-orig:2": ‹∀α (φ → ψ{α}) → (φ → ∀α ψ{α})›
453  by (metis "→I" GEN "vdash-properties:6" "∀E"(4))
454AOT_theorem "cqt-orig:3": ‹∀α φ{α} → φ{α}› using "cqt-orig:1[const_var]" .
455
456(* TODO: work out difference to GEN *)
457AOT_theorem universal: assumes ‹for arbitrary β: φ{β}› shows ‹∀α φ{α}›
458  using GEN assms .
459lemmas "∀I" = universal
460
461(* Generalized mechanism for "∀I" followed by ∀E *)
462ML‹
463fun get_instantiated_allI ctxt varname thm = let
464val trm = Thm.concl_of thm
465val trm = case trm of (@{const Trueprop} $ (@{const AOT_model_valid_in} $ _ $ x)) => x
466                      | _ => raise Term.TERM ("Expected simple theorem.", [trm])
467fun extractVars (Const (const_name‹AOT_term_of_var›, _) $ Var v) =
468    (if fst (fst v) = fst varname then [Var v] else []) (* TODO: care about the index? *)
469  | extractVars (t1 $ t2) = extractVars t1 @ extractVars t2
470  | extractVars (Abs (_, _, t)) = extractVars t
471  | extractVars _ = []
472val vars = extractVars trm
473val vars = fold Term.add_vars vars []
474val var = hd vars
475val trmty = case (snd var) of (Type (type_name‹AOT_var›, [t])) => (t)
476              | _ => raise Term.TYPE ("Expected variable type.", [snd var], [Var var])
477val trm = Abs (Term.string_of_vname (fst var), trmty, Term.abstract_over (
478      Const (const_name‹AOT_term_of_var›, Type ("fun", [snd var, trmty]))
479       $ Var var, trm))
480val trm = Thm.cterm_of (Context.proof_of ctxt) trm
481val ty = hd (Term.add_tvars (Thm.prop_of @{thm "∀I"}) [])
482val typ = Thm.ctyp_of (Context.proof_of ctxt) trmty
483val allthm = Drule.instantiate_normalize ([(ty, typ)],[]) @{thm "∀I"}
484val phi = hd (Term.add_vars (Thm.prop_of allthm) [])
485val allthm = Drule.instantiate_normalize ([],[(phi,trm)]) allthm
486in
487allthm
488end
489›
490
491attribute_setup "∀I" =
492  ‹Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
493  (fn ctxt => fn thm => fold (fn arg => fn thm => thm RS get_instantiated_allI ctxt arg thm) args thm))›
494  "Quantify over a variable in a theorem using GEN."
495
496attribute_setup "unvarify" =
497  ‹Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
498  (fn ctxt => fn thm =>
499    let
500    val thm = fold (fn arg => fn thm => thm RS get_instantiated_allI ctxt arg thm) args thm
501    val thm = fold (fn _ => fn thm => thm RS @{thm "∀E"(1)}) args thm
502    in
503     thm
504    end))›
505  "Generalize a statement about variables to a statement about denoting terms."
506
507(* TODO: rereplace-lem does not apply to the embedding *)
508
509AOT_theorem "cqt-basic:1": ‹∀α∀β φ{α,β} ≡ ∀β∀α φ{α,β}›
510  by (metis "≡I" "∀E"(2) "∀I" "→I")
511
512AOT_theorem "cqt-basic:2": ‹∀α(φ{α} ≡ ψ{α}) ≡ (∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α}))›
513proof (rule "≡I"; rule "→I")
514  AOT_assume ‹∀α(φ{α} ≡ ψ{α})›
515  AOT_hence ‹φ{α} ≡ ψ{α}› for α using "∀E"(2) by blast
516  AOT_hence ‹φ{α} → ψ{α}› and ‹ψ{α} → φ{α}› for α
517    using "≡E"(1,2) "→I" by blast+
518  AOT_thus ‹∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α})›
519    by (auto intro: "&I" "∀I")
520next
521  AOT_assume ‹∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α})›
522  AOT_hence ‹φ{α} → ψ{α}› and ‹ψ{α} → φ{α}› for α
523    using "∀E"(2) "&E" by blast+
524  AOT_hence ‹φ{α} ≡ ψ{α}› for α
525    using "≡I" by blast
526  AOT_thus ‹∀α(φ{α} ≡ ψ{α})› by (auto intro: "∀I")
527qed
528
529AOT_theorem "cqt-basic:3": ‹∀α(φ{α} ≡ ψ{α}) → (∀α φ{α} ≡ ∀α ψ{α})›
530proof(rule "→I")
531  AOT_assume ‹∀α(φ{α} ≡ ψ{α})›
532  AOT_hence 1: ‹φ{α} ≡ ψ{α}› for α using "∀E"(2) by blast
533  {
534    AOT_assume ‹∀α φ{α}›
535    AOT_hence ‹∀α ψ{α}› using 1 "∀I" "∀E"(4) "≡E" by metis
536  }
537  moreover {
538    AOT_assume ‹∀α ψ{α}›
539    AOT_hence ‹∀α φ{α}› using 1 "∀I" "∀E"(4) "≡E" by metis
540  }
541  ultimately AOT_show ‹∀α φ{α} ≡ ∀α ψ{α}›
542    using "≡I" "→I" by auto
543qed
544
545AOT_theorem "cqt-basic:4": ‹∀α(φ{α} & ψ{α}) → (∀α φ{α} & ∀α ψ{α})›
546proof(rule "→I")
547  AOT_assume 0: ‹∀α(φ{α} & ψ{α})›
548  AOT_have ‹φ{α}› and ‹ψ{α}› for α using "∀E"(2) 0 "&E" by blast+
549  AOT_thus ‹∀α φ{α} & ∀α ψ{α}›
550    by (auto intro: "∀I" "&I")
551qed
552
553AOT_theorem "cqt-basic:5": ‹(∀α1...∀αn(φ{α1...αn})) → φ{α1...αn}›
554  using "cqt-orig:3" by blast
555
556AOT_theorem "cqt-basic:6": ‹∀α∀α φ{α} ≡ ∀α φ{α}›
557  by (meson "≡I" "→I" GEN "cqt-orig:1[const_var]")
558
559AOT_theorem "cqt-basic:7": ‹(φ → ∀α ψ{α}) ≡ ∀α(φ → ψ{α})›
560  by (metis "→I" "vdash-properties:6" "rule-ui:3" "≡I" GEN)
561
562AOT_theorem "cqt-basic:8": ‹(∀α φ{α} ∨ ∀α ψ{α}) → ∀α (φ{α} ∨ ψ{α})›
563  by (simp add: "∨I"(3) "→I" GEN "cqt-orig:1[const_var]")
564
565AOT_theorem "cqt-basic:9": ‹(∀α (φ{α} → ψ{α}) & ∀α (ψ{α} → χ{α})) → ∀α(φ{α} → χ{α})›
566proof -
567  {
568    AOT_assume ‹∀α (φ{α} → ψ{α})›
569    moreover AOT_assume ‹∀α (ψ{α} → χ{α})›
570    ultimately AOT_have ‹φ{α} → ψ{α}› and ‹ψ{α} → χ{α}› for α using "∀E" by blast+
571    AOT_hence ‹φ{α} → χ{α}› for α by (metis "→E" "→I")
572    AOT_hence ‹∀α(φ{α} → χ{α})› using "∀I" by fast
573  }
574  thus ?thesis using "&I" "→I" "&E" by meson
575qed
576
577AOT_theorem "cqt-basic:10": ‹(∀α(φ{α} ≡ ψ{α}) & ∀α(ψ{α} ≡ χ{α})) → ∀α (φ{α} ≡ χ{α})›
578proof(rule "→I"; rule "∀I")
579  fix β
580  AOT_assume ‹∀α(φ{α} ≡ ψ{α}) & ∀α(ψ{α} ≡ χ{α})›
581  AOT_hence ‹φ{β} ≡ ψ{β}› and ‹ψ{β} ≡ χ{β}› using "&E" "∀E" by blast+
582  AOT_thus ‹φ{β} ≡ χ{β}› using "≡I" "≡E" by blast
583qed
584
585AOT_theorem "cqt-basic:11": ‹∀α(φ{α} ≡ ψ{α}) ≡ ∀α (ψ{α} ≡ φ{α})›
586proof (rule "≡I"; rule "→I")
587  AOT_assume 0: ‹∀α(φ{α} ≡ ψ{α})›
588  {
589    fix α
590    AOT_have ‹φ{α} ≡ ψ{α}› using 0 "∀E" by blast
591    AOT_hence ‹ψ{α} ≡ φ{α}› using "≡I" "≡E" "→I" "→E" by metis
592  }
593  AOT_thus ‹∀α(ψ{α} ≡ φ{α})› using "∀I" by fast
594next
595  AOT_assume 0: ‹∀α(ψ{α} ≡ φ{α})›
596  {
597    fix α
598    AOT_have ‹ψ{α} ≡ φ{α}› using 0 "∀E" by blast
599    AOT_hence ‹φ{α} ≡ ψ{α}› using "≡I" "≡E" "→I" "→E" by metis
600  }
601  AOT_thus ‹∀α(φ{α} ≡ ψ{α})› using "∀I" by fast
602qed
603
604AOT_theorem "cqt-basic:12": ‹∀α φ{α} → ∀α (ψ{α} → φ{α})›
605  by (simp add: "∀E"(2) "→I" GEN)
606
607AOT_theorem "cqt-basic:13": ‹∀α φ{α} ≡ ∀β φ{β}›
608  using "≡I" "→I" by blast
609
610AOT_theorem "cqt-basic:14": ‹(∀α1...∀αn (φ{α1...αn} → ψ{α1...αn})) → ((∀α1...∀αn φ{α1...αn}) → (∀α1...∀αn ψ{α1...αn}))›
611  using "cqt:3"[axiom_inst] by auto
612
613AOT_theorem "cqt-basic:15": ‹(∀α1...∀αn (φ → ψ{α1...αn})) → (φ → (∀α1...∀αn ψ{α1...αn}))›
614  using "cqt-orig:2" by auto
615
616(* TODO: once more the same in the embedding... need to distinguish these better *)
617AOT_theorem "universal-cor": assumes ‹for arbitrary β: φ{β}›  shows ‹∀α φ{α}›
618  using GEN assms .
619
620AOT_theorem "existential:1": assumes ‹φ{τ}› and ‹τ↓› shows ‹∃α φ{α}›
621proof(rule "raa-cor:1")
622  AOT_assume ‹¬∃α φ{α}›
623  AOT_hence ‹∀α ¬φ{α}›
624    using "≡dfI" "conventions:4" RAA "&I" by blast
625  AOT_hence ‹¬φ{τ}› using assms(2) "∀E"(1) "→E" by blast
626  AOT_thus ‹φ{τ} & ¬φ{τ}› using assms(1) "&I" by blast
627qed
628
629AOT_theorem "existential:2[const_var]": assumes ‹φ{β}› shows ‹∃α φ{α}›
630  using "existential:1" "cqt:2[const_var]"[axiom_inst] assms by blast
631
632AOT_theorem "existential:2[lambda]":
633  assumes ‹φ{[λν1...νn ψ{ν1...νn}]}› and ‹INSTANCE_OF_CQT_2(ψ)›
634  shows ‹∃α φ{α}›
635  using "existential:1" "cqt:2[lambda]"[axiom_inst] assms by blast
636lemmas "∃I" = "existential:1" "existential:2[const_var]" "existential:2[lambda]" 
637
638AOT_theorem "instantiation":
639  assumes ‹for arbitrary β: φ{β} ⊢ ψ› and ‹∃α φ{α}›
640  shows ‹ψ›
641  by (metis (no_types, lifting) "≡dfE" GEN "raa-cor:3" "conventions:4" assms)
642lemmas "∃E" = "instantiation"
643
644AOT_theorem "cqt-further:1": ‹∀α φ{α} → ∃α φ{α}›
645  using "∀E"(4) "∃I"(2) "→I" by metis
646
647AOT_theorem "cqt-further:2": ‹¬∀α φ{α} → ∃α ¬φ{α}›
648  using "∀I" "∃I"(2) "→I" RAA by metis
649
650AOT_theorem "cqt-further:3": ‹∀α φ{α} ≡ ¬∃α ¬φ{α}›
651  using "∀E"(4) "∃E" "→I" RAA
652  by (metis "cqt-further:2" "≡I" "modus-tollens:1")
653
654AOT_theorem "cqt-further:4": ‹¬∃α φ{α} → ∀α ¬φ{α}›
655  using "∀I" "∃I"(2)"→I" RAA by metis
656
657AOT_theorem "cqt-further:5": ‹∃α (φ{α} & ψ{α}) → (∃α φ{α} & ∃α ψ{α})›
658  by (metis (no_types, lifting) "&E" "&I" "∃E" "∃I"(2) "→I")
659
660AOT_theorem "cqt-further:6": ‹∃α (φ{α} ∨ ψ{α}) → (∃α φ{α} ∨ ∃α ψ{α})›
661  by (metis (mono_tags, lifting) "∃E" "∃I"(2) "∨E"(3) "∨I"(1, 2) "→I" RAA(2))
662
663AOT_theorem "cqt-further:7": ‹∃α φ{α} ≡ ∃β φ{β}› (* TODO: vacuous in the embedding *)
664  by (simp add: "oth-class-taut:3:a")
665
666AOT_theorem "cqt-further:8": ‹(∀α φ{α} & ∀α ψ{α}) → ∀α (φ{α} ≡ ψ{α})›
667  by (metis (mono_tags, lifting) "&E" "≡I" "∀E"(2) "→I" GEN)
668
669AOT_theorem "cqt-further:9": ‹(¬∃α φ{α} & ¬∃α ψ{α}) → ∀α (φ{α} ≡ ψ{α})›
670  by (metis (mono_tags, lifting) "&E" "≡I" "∃I"(2) "→I" GEN "raa-cor:4")
671
672AOT_theorem "cqt-further:10": ‹(∃α φ{α} & ¬∃α ψ{α}) → ¬∀α (φ{α} ≡ ψ{α})›
673proof(rule "→I"; rule "raa-cor:2")
674  AOT_assume 0: ‹∃α φ{α} & ¬∃α ψ{α}›
675  then AOT_obtain α where ‹φ{α}› using "∃E" "&E"(1) by metis
676  moreover AOT_assume ‹∀α (φ{α} ≡ ψ{α})›
677  ultimately AOT_have ‹ψ{α}› using "∀E"(4) "≡E"(1) by blast
678  AOT_hence ‹∃α ψ{α}› using "∃I" by blast
679  AOT_thus ‹∃α ψ{α} & ¬∃α ψ{α}› using 0 "&E"(2) "&I" by blast
680qed
681
682AOT_theorem "cqt-further:11": ‹∃α∃β φ{α,β} ≡ ∃β∃α φ{α,β}›
683  using "≡I" "→I" "∃I"(2) "∃E" by metis
684
685AOT_theorem "log-prop-prop:1": ‹[λ φ]↓›
686  using "cqt:2[lambda0]"[axiom_inst] by auto
687
688AOT_theorem "log-prop-prop:2": ‹φ↓›
689  by (rule "≡dfI"[OF "existence:3"]) "cqt:2[lambda]"
690
691AOT_theorem "exist-nec": ‹τ↓ → □τ↓›
692proof -
693  AOT_have ‹∀β □β↓›
694    by (simp add: GEN RN "cqt:2[const_var]"[axiom_inst])
695  AOT_thus ‹τ↓ → □τ↓›
696    using "cqt:1"[axiom_inst] "→E" by blast
697qed
698
699(* TODO: replace this mechanism by a "proof by types" command *)
700class AOT_Term_id = AOT_Term +
701  assumes "t=t-proper:1"[AOT]: ‹[v ⊨ τ = τ' → τ↓]›
702      and "t=t-proper:2"[AOT]: ‹[v ⊨ τ = τ' → τ'↓]›
703
704instance κ :: AOT_Term_id
705proof
706  AOT_modally_strict {
707    AOT_show ‹κ = κ' → κ↓› for κ κ'
708    proof(rule "→I")
709      AOT_assume ‹κ = κ'›
710      AOT_hence ‹O!κ ∨ A!κ›
711        by (rule "∨I"(3)[OF "≡dfE"[OF "identity:1"]])
712           (meson "→I" "∨I"(1) "&E"(1))+
713      AOT_thus ‹κ↓›
714        by (rule "∨E"(1))
715           (metis "cqt:5:a"[axiom_inst] "→I" "→E" "&E"(2))+
716    qed
717  }
718next
719  AOT_modally_strict {
720    AOT_show ‹κ = κ' → κ'↓› for κ κ'
721    proof(rule "→I")
722      AOT_assume ‹κ = κ'›
723      AOT_hence ‹O!κ' ∨ A!κ'›
724        by (rule "∨I"(3)[OF "≡dfE"[OF "identity:1"]])
725           (meson "→I" "∨I" "&E")+
726      AOT_thus ‹κ'↓›
727        by (rule "∨E"(1))
728           (metis "cqt:5:a"[axiom_inst] "→I" "→E" "&E"(2))+
729    qed
730  }
731qed
732
733instance rel :: (AOT_κs) AOT_Term_id
734proof
735  AOT_modally_strict {
736    AOT_show ‹Π = Π' → Π↓› for Π Π' :: ‹<'a>› (* TODO: how to get rid of the fixes? *)
737    proof(rule "→I")
738      AOT_assume ‹Π = Π'›
739      AOT_thus ‹Π↓› using "≡dfE"[OF "identity:3"[of Π Π']] "&E" by blast
740    qed
741  }
742next
743  AOT_modally_strict {
744    AOT_show ‹Π = Π' → Π'↓› for Π Π' :: ‹<'a>› (* TODO: how to get rid of the fixes? *)
745    proof(rule "→I")
746      AOT_assume ‹Π = Π'›
747      AOT_thus ‹Π'↓› using "≡dfE"[OF "identity:3"[of Π Π']] "&E" by blast
748    qed
749  }
750qed
751
752instance 𝗈 :: AOT_Term_id
753proof
754  AOT_modally_strict {
755    fix φ ψ
756    AOT_show ‹φ = ψ → φ↓›
757    proof(rule "→I")
758      AOT_assume ‹φ = ψ›
759      AOT_thus ‹φ↓› using "≡dfE"[OF "identity:4"[of φ ψ]] "&E" by blast
760    qed
761  }
762next
763  AOT_modally_strict {
764    fix φ ψ
765    AOT_show ‹φ = ψ → ψ↓›
766    proof(rule "→I")
767      AOT_assume ‹φ = ψ›
768      AOT_thus ‹ψ↓› using "≡dfE"[OF "identity:4"[of φ ψ]] "&E" by blast
769    qed
770  }
771qed
772
773instance prod :: (AOT_Term_id, AOT_Term_id) AOT_Term_id
774proof
775  AOT_modally_strict {
776    fix τ τ' :: ‹'a×'b›
777    AOT_show ‹τ = τ' → τ↓›
778    proof (induct τ; induct τ'; rule "→I")
779      fix τ1 τ1' :: 'a and τ2  τ2' :: 'b
780      AOT_assume ‹«(τ1, τ2)» = «(τ1', τ2')»›
781      AOT_hence ‹(τ1 = τ1') & (τ2 = τ2')› by (metis "≡dfE" tuple_identity_1)
782      AOT_hence ‹τ1↓› and ‹τ2↓› using "t=t-proper:1" "&E" "vdash-properties:10" by blast+
783      AOT_thus ‹«(τ1, τ2)»↓› by (metis "≡dfI" "&I" tuple_denotes)
784    qed
785  }
786next
787  AOT_modally_strict {
788    fix τ τ' :: ‹'a×'b›
789    AOT_show ‹τ = τ' → τ'↓›
790    proof (induct τ; induct τ'; rule "→I")
791      fix τ1 τ1' :: 'a and τ2  τ2' :: 'b
792      AOT_assume ‹«(τ1, τ2)» = «(τ1', τ2')»›
793      AOT_hence ‹(τ1 = τ1') & (τ2 = τ2')› by (metis "≡dfE" tuple_identity_1)
794      AOT_hence ‹τ1'↓› and ‹τ2'↓› using "t=t-proper:2" "&E" "vdash-properties:10" by blast+
795      AOT_thus ‹«(τ1', τ2')»↓› by (metis "≡dfI" "&I" tuple_denotes)
796    qed
797  }
798qed
799
800(* TODO: this is the end of the "proof by types" and makes the results available on new theorems *)
801AOT_register_type_constraints
802  Term: ‹_::AOT_Term_id› ‹_::AOT_Term_id›
803AOT_register_type_constraints
804  Individual: ‹κ› ‹_::{AOT_κs, AOT_Term_id}›
805AOT_register_type_constraints
806  Relation: ‹<_::{AOT_κs, AOT_Term_id}>›
807
808AOT_theorem "id-rel-nec-equiv:1": ‹Π = Π' → □∀x1...∀xn ([Π]x1...xn ≡ [Π']x1...xn)›
809proof(rule "→I")
810  AOT_assume assumption: ‹Π = Π'›
811  AOT_hence ‹Π↓› and ‹Π'↓›
812    using "t=t-proper:1" "t=t-proper:2" MP by blast+
813  moreover AOT_have ‹∀F∀G (F = G → ((□∀x1...∀xn ([F]x1...xn ≡ [F]x1...xn)) → □∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)))›
814    apply (rule GEN)+ using "l-identity"[axiom_inst] by force
815  ultimately AOT_have ‹Π = Π' → ((□∀x1...∀xn ([Π]x1...xn ≡ [Π]x1...xn)) → □∀x1...∀xn ([Π]x1...xn ≡ [Π']x1...xn))›
816    using "∀E"(1) by blast
817  AOT_hence ‹(□∀x1...∀xn ([Π]x1...xn ≡ [Π]x1...xn)) → □∀x1...∀xn ([Π]x1...xn ≡ [Π']x1...xn)›
818    using assumption "→E" by blast
819  moreover AOT_have ‹□∀x1...∀xn ([Π]x1...xn ≡ [Π]x1...xn)›
820    by (simp add: RN "oth-class-taut:3:a" "universal-cor")
821  ultimately AOT_show ‹□∀x1...∀xn ([Π]x1...xn ≡ [Π']x1...xn)›
822    using "→E" by blast
823qed
824
825AOT_theorem "id-rel-nec-equiv:2": ‹φ = ψ → □(φ ≡ ψ)›
826proof(rule "→I")
827  AOT_assume assumption: ‹φ = ψ›
828  AOT_hence ‹φ↓› and ‹ψ↓›
829    using "t=t-proper:1" "t=t-proper:2" MP by blast+
830  moreover AOT_have ‹∀p∀q (p = q → ((□(p ≡ p) → □(p ≡ q))))›
831    apply (rule GEN)+ using "l-identity"[axiom_inst] by force
832  ultimately AOT_have ‹φ = ψ → (□(φ ≡ φ) → □(φ ≡ ψ))›
833    using "∀E"(1) by blast
834  AOT_hence ‹□(φ ≡ φ) → □(φ ≡ ψ)›
835    using assumption "→E" by blast
836  moreover AOT_have ‹□(φ ≡ φ)›
837    by (simp add: RN "oth-class-taut:3:a" "universal-cor")
838  ultimately AOT_show ‹□(φ ≡ ψ)›
839    using "→E" by blast
840qed
841
842AOT_theorem "rule=E": assumes ‹φ{τ}› and ‹τ = σ› shows ‹φ{σ}›
843proof -
844  AOT_have ‹τ↓› and ‹σ↓› using assms(2) "t=t-proper:1" "t=t-proper:2" "→E" by blast+
845  moreover AOT_have ‹∀α∀β(α = β → (φ{α} → φ{β}))›
846    apply (rule GEN)+ using "l-identity"[axiom_inst] by blast
847  ultimately AOT_have ‹τ = σ → (φ{τ} → φ{σ})›
848    using "∀E"(1) by blast
849  AOT_thus ‹φ{σ}› using assms "→E" by blast
850qed
851
852AOT_theorem "propositions-lemma:1": ‹[λ φ] = φ›
853proof -
854  AOT_have ‹φ↓› by (simp add: "log-prop-prop:2")
855  moreover AOT_have ‹∀p [λ p] = p› using "lambda-predicates:3[zero]"[axiom_inst] "∀I" by fast
856  ultimately AOT_show ‹[λ φ] = φ›
857    using "∀E" by blast
858qed
859
860AOT_theorem "propositions-lemma:2": ‹[λ φ] ≡ φ›
861proof -
862  AOT_have ‹[λ φ] ≡ [λ φ]› by (simp add: "oth-class-taut:3:a")
863  AOT_thus ‹[λ φ] ≡ φ› using "propositions-lemma:1" "rule=E" by blast
864qed
865
866(* propositions-lemma:3 through propositions-lemma:5 do not apply *)
867
868AOT_theorem "propositions-lemma:6": ‹(φ ≡ ψ) ≡ ([λ φ] ≡ [λ ψ])›
869  by (metis "≡E"(1) "≡E"(5) "Associativity of ≡" "propositions-lemma:2")
870
871(* dr-alphabetic-rules does not apply *)
872
873AOT_theorem "oa-exist:1": ‹O!↓›
874proof -
875  AOT_have ‹[λx ◇[E!]x]↓› by "cqt:2[lambda]"
876  AOT_hence 1: ‹O! = [λx ◇[E!]x]› using "df-rules-terms[4]"[OF "oa:1", THEN "&E"(1)] "→E" by blast
877  AOT_show ‹O!↓› using "t=t-proper:1"[THEN "→E", OF 1] by simp
878qed
879
880AOT_theorem "oa-exist:2": ‹A!↓›
881proof -
882  AOT_have ‹[λx ¬◇[E!]x]↓› by "cqt:2[lambda]"
883  AOT_hence 1: ‹A! = [λx ¬◇[E!]x]› using "df-rules-terms[4]"[OF "oa:2", THEN "&E"(1)] "→E" by blast
884  AOT_show ‹A!↓› using "t=t-proper:1"[THEN "→E", OF 1] by simp
885qed
886
887AOT_theorem "oa-exist:3": ‹O!x ∨ A!x›
888proof(rule "raa-cor:1")
889  AOT_assume ‹¬(O!x ∨ A!x)›
890  AOT_hence A: ‹¬O!x› and B: ‹¬A!x›
891    using "Disjunction Addition"(1) "modus-tollens:1" "∨I"(2) "raa-cor:5" by blast+
892  AOT_have C: ‹O! = [λx ◇[E!]x]›
893    by (rule "df-rules-terms[4]"[OF "oa:1", THEN "&E"(1), THEN "→E"]) "cqt:2[lambda]"
894  AOT_have D: ‹A! = [λx ¬◇[E!]x]›
895    by (rule "df-rules-terms[4]"[OF "oa:2", THEN "&E"(1), THEN "→E"]) "cqt:2[lambda]"
896  AOT_have E: ‹¬[λx ◇[E!]x]x›
897    using A C "rule=E" by fast
898  AOT_have F: ‹¬[λx ¬◇[E!]x]x›
899    using B D "rule=E" by fast
900  AOT_have G: ‹[λx ◇[E!]x]x ≡ ◇[E!]x›
901    by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
902  AOT_have H: ‹[λx ¬◇[E!]x]x ≡ ¬◇[E!]x›
903    by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
904  AOT_show ‹¬◇[E!]x & ¬¬◇[E!]x› using G E "≡E" H F "≡E" "&I" by metis
905qed
906
907AOT_theorem "p-identity-thm2:1": ‹F = G ≡ □∀x(x[F] ≡ x[G])›
908proof -
909  AOT_have ‹F = G ≡ F↓ & G↓ & □∀x(x[F] ≡ x[G])›
910    using "identity:2" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
911  moreover AOT_have ‹F↓› and ‹G↓›
912    by (auto simp: "cqt:2[const_var]"[axiom_inst])
913  ultimately AOT_show ‹F = G ≡ □∀x(x[F] ≡ x[G])›
914    using "≡S"(1) "&I" by blast
915qed
916
917AOT_theorem "p-identity-thm2:2[2]": ‹F = G ≡ ∀y1([λx [F]xy1] = [λx [G]xy1] & [λx [F]y1x] = [λx [G]y1x])›
918proof -
919  AOT_have ‹F = G ≡ F↓ & G↓ & ∀y1([λx [F]xy1] = [λx [G]xy1] & [λx [F]y1x] = [λx [G]y1x])›
920    using "identity:3[2]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
921  moreover AOT_have ‹F↓› and ‹G↓›
922    by (auto simp: "cqt:2[const_var]"[axiom_inst])
923  ultimately show ?thesis
924    using "≡S"(1) "&I" by blast
925qed
926    
927AOT_theorem "p-identity-thm2:2[3]": ‹F = G ≡ ∀y1∀y2([λx [F]xy1y2] = [λx [G]xy1y2] & [λx [F]y1xy2] = [λx [G]y1xy2] & [λx [F]y1y2x] = [λx [G]y1y2x])›
928proof -
929  AOT_have ‹F = G ≡ F↓ & G↓ & ∀y1∀y2([λx [F]xy1y2] = [λx [G]xy1y2] & [λx [F]y1xy2] = [λx [G]y1xy2] & [λx [F]y1y2x] = [λx [G]y1y2x])›
930    using "identity:3[3]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
931  moreover AOT_have ‹F↓› and ‹G↓›
932    by (auto simp: "cqt:2[const_var]"[axiom_inst])
933  ultimately show ?thesis
934    using "≡S"(1) "&I" by blast
935qed
936
937AOT_theorem "p-identity-thm2:2[4]": ‹F = G ≡ ∀y1∀y2∀y3([λx [F]xy1y2y3] = [λx [G]xy1y2y3] & [λx [F]y1xy2y3] = [λx [G]y1xy2y3] & [λx [F]y1y2xy3] = [λx [G]y1y2xy3] & [λx [F]y1y2y3x] = [λx [G]y1y2y3x])›
938proof -
939  AOT_have ‹F = G ≡ F↓ & G↓ & ∀y1∀y2∀y3([λx [F]xy1y2y3] = [λx [G]xy1y2y3] & [λx [F]y1xy2y3] = [λx [G]y1xy2y3] & [λx [F]y1y2xy3] = [λx [G]y1y2xy3] & [λx [F]y1y2y3x] = [λx [G]y1y2y3x])›
940    using "identity:3[4]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
941  moreover AOT_have ‹F↓› and ‹G↓›
942    by (auto simp: "cqt:2[const_var]"[axiom_inst])
943  ultimately show ?thesis
944    using "≡S"(1) "&I" by blast
945qed
946
947AOT_theorem "p-identity-thm2:2":
948  ‹F = G ≡ ∀x1...∀xn «AOT_sem_proj_id x1xn (λ τ . «[F]τ») (λ τ . «[G]τ»)»›
949proof -
950  AOT_have ‹F = G ≡ F↓ & G↓ & ∀x1...∀xn «AOT_sem_proj_id x1xn (λ τ . «[F]τ») (λ τ . «[G]τ»)»›
951    using "identity:3" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
952  moreover AOT_have ‹F↓› and ‹G↓›
953    by (auto simp: "cqt:2[const_var]"[axiom_inst])
954  ultimately show ?thesis
955    using "≡S"(1) "&I" by blast
956qed
957
958AOT_theorem "p-identity-thm2:3":
959  ‹p = q ≡ [λx p] = [λx q]›
960proof -
961  AOT_have ‹p = q ≡ p↓ & q↓ & [λx p] = [λx q]›
962    using "identity:4" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
963  moreover AOT_have ‹p↓› and ‹q↓›
964    by (auto simp: "cqt:2[const_var]"[axiom_inst])
965  ultimately show ?thesis
966    using "≡S"(1) "&I" by blast
967qed
968
969class AOT_Term_id_2 = AOT_Term_id + assumes "id-eq:1": ‹[v ⊨ α = α]›
970
971instance κ :: AOT_Term_id_2
972proof
973  AOT_modally_strict {
974    fix x
975    {
976      AOT_assume ‹O!x›
977      moreover AOT_have ‹□∀F([F]x ≡ [F]x)›
978        using RN GEN "oth-class-taut:3:a" by fast
979      ultimately AOT_have ‹O!x & O!x & □∀F([F]x ≡ [F]x)› using "&I" by simp
980    }
981    moreover {
982      AOT_assume ‹A!x›
983      moreover AOT_have ‹□∀F(x[F] ≡ x[F])›
984        using RN GEN "oth-class-taut:3:a" by fast
985      ultimately AOT_have ‹A!x & A!x & □∀F(x[F] ≡ x[F])› using "&I" by simp
986    }
987    ultimately AOT_have ‹(O!x & O!x & □∀F([F]x ≡ [F]x)) ∨ (A!x & A!x & □∀F(x[F] ≡ x[F]))›
988      using "oa-exist:3" "∨I"(1) "∨I"(2) "∨E"(3) "raa-cor:1" by blast
989    AOT_thus ‹x = x›
990      using "identity:1"[THEN "df-rules-formulas[4]"] "→E" by blast
991  }
992qed
993
994instance rel :: ("{AOT_κs,AOT_Term_id_2}") AOT_Term_id_2
995proof
996  AOT_modally_strict {
997    fix F :: "<'a> AOT_var"
998    AOT_have 0: ‹[λx1...xn [F]x1...xn] = F›
999      by (simp add: "lambda-predicates:3"[axiom_inst])
1000    AOT_have ‹[λx1...xn [F]x1...xn]↓›
1001      by "cqt:2[lambda]"
1002    AOT_hence ‹[λx1...xn [F]x1...xn] = [λx1...xn [F]x1...xn]›
1003      using "lambda-predicates:1"[axiom_inst] "→E" by blast
1004    AOT_show ‹F = F› using "rule=E" 0 by force 
1005  }
1006qed
1007
1008instance 𝗈 :: AOT_Term_id_2
1009proof
1010  AOT_modally_strict {
1011    fix p
1012    AOT_have 0: ‹[λ p] = p›
1013      by (simp add: "lambda-predicates:3[zero]"[axiom_inst])
1014    AOT_have ‹[λ p]↓›
1015      by (rule "cqt:2[lambda0]"[axiom_inst])
1016    AOT_hence ‹[λ p] = [λ p]›
1017      using "lambda-predicates:1[zero]"[axiom_inst] "→E" by blast
1018    AOT_show ‹p = p› using "rule=E" 0 by force
1019  }
1020qed
1021
1022instance prod :: (AOT_Term_id_2, AOT_Term_id_2) AOT_Term_id_2
1023proof
1024  AOT_modally_strict {
1025    fix α :: ‹('a×'b) AOT_var›
1026    AOT_show ‹α = α›
1027    proof (induct)
1028      AOT_show ‹τ = τ› if ‹τ↓› for τ :: ‹'a×'b›
1029        using that
1030      proof (induct τ)
1031        fix τ1 :: 'a and τ2 :: 'b
1032        AOT_assume ‹«(τ1,τ2)»↓›
1033        AOT_hence ‹τ1↓› and ‹τ2↓› using "≡dfE" "&E" tuple_denotes by blast+
1034        AOT_hence ‹τ1 = τ1› and ‹τ2 = τ2› using "id-eq:1"[unvarify α] by blast+
1035        AOT_thus ‹«(τ1, τ2)» = «(τ1, τ2)»› by (metis "≡dfI" "&I" tuple_identity_1)
1036      qed
1037    qed
1038  }
1039qed
1040
1041AOT_register_type_constraints
1042  Term: ‹_::AOT_Term_id_2› ‹_::AOT_Term_id_2›
1043AOT_register_type_constraints
1044  Individual: ‹κ› ‹_::{AOT_κs, AOT_Term_id_2}›
1045AOT_register_type_constraints
1046  Relation: ‹<_::{AOT_κs, AOT_Term_id_2}>›
1047
1048(* TODO: Interestingly, this doesn't depend on "id-eq:1" at all! *)
1049AOT_theorem "id-eq:2": ‹α = β → β = α›
1050(*
1051  TODO: look at this proof generated using:
1052        including AOT_no_atp sledgehammer[isar_proofs = true]
1053proof -
1054  have "(∃φ. [v ⊨ ~β = α → ~φ] ∧ [v ⊨ α = β → φ]) ∨ (∃φ. ¬ [v ⊨ φ{α} → φ{β}])"
1055    by meson
1056  then show ?thesis
1057    by (meson "contraposition:2" "Hypothetical Syllogism" "deduction-theorem" l_"identity:1" "useful-tautologies:1")
1058qed
1059*)
1060(*  by (meson "rule=E" "deduction-theorem") *)
1061proof (rule "→I")
1062  AOT_assume ‹α = β›
1063  moreover AOT_have ‹β = β› using calculation "rule=E"[of _ "λ τ . «τ = β»" "AOT_term_of_var α" "AOT_term_of_var β"] by blast
1064  moreover AOT_have ‹α = α → α = α› using "if-p-then-p" by blast
1065  ultimately AOT_show ‹β = α›
1066    using "→E" "→I" "rule=E"[of _ "λ τ . «(τ = τ) → (τ = α)»" "AOT_term_of_var α" "AOT_term_of_var β"] by blast
1067qed
1068
1069AOT_theorem "id-eq:3": ‹α = β & β = γ → α = γ›
1070  using "rule=E" "→I" "&E" by blast
1071
1072AOT_theorem "id-eq:4": ‹α = β ≡ ∀γ (α = γ ≡ β = γ)›
1073proof (rule "≡I"; rule "→I")
1074  AOT_assume 0: ‹α = β›
1075  AOT_hence 1: ‹β = α› using "id-eq:2" "→E" by blast
1076  AOT_show ‹∀γ (α = γ ≡ β = γ)›
1077    by (rule GEN) (metis "≡I" "→I" 0 "1" "rule=E")
1078next
1079  AOT_assume ‹∀γ (α = γ ≡ β = γ)›
1080  AOT_hence ‹α = α ≡ β = α› using "∀E"(2) by blast
1081  AOT_hence ‹α = α → β = α› using "≡E"(1) "→I" by blast
1082  AOT_hence ‹β = α› using "id-eq:1" "→E" by blast
1083  AOT_thus ‹α = β› using "id-eq:2" "→E" by blast
1084qed
1085
1086AOT_theorem "rule=I:1": assumes ‹τ↓› shows ‹τ = τ›
1087proof -
1088  AOT_have ‹∀α (α = α)›
1089    by (rule GEN) (metis "id-eq:1")
1090  AOT_thus ‹τ = τ› using assms "∀E" by blast
1091qed
1092
1093AOT_theorem "rule=I:2[const_var]": "α = α"
1094  using "id-eq:1".
1095
1096AOT_theorem "rule=I:2[lambda]": assumes ‹INSTANCE_OF_CQT_2(φ)› shows "[λν1...νn φ{ν1...νn}] = [λν1...νn φ{ν1...νn}]"
1097proof -
1098  AOT_have ‹∀α (α = α)›
1099    by (rule GEN) (metis "id-eq:1")
1100  moreover AOT_have ‹[λν1...νn φ{ν1...νn}]↓› using assms by (rule "cqt:2[lambda]"[axiom_inst])
1101  ultimately AOT_show ‹[λν1...νn φ{ν1...νn}] = [λν1...νn φ{ν1...νn}]› using assms "∀E" by blast
1102qed
1103
1104lemmas "=I" = "rule=I:1" "rule=I:2[const_var]" "rule=I:2[lambda]"
1105
1106AOT_theorem "rule-id-def:1":
1107  assumes ‹τ{α1...αn} =df σ{α1...αn}› and ‹σ{τ1...τn}↓›
1108  shows ‹τ{τ1...τn} = σ{τ1...τn}›
1109proof -
1110  AOT_have ‹σ{τ1...τn}↓ → τ{τ1...τn} = σ{τ1...τn}›
1111    using "df-rules-terms[3]" assms(1) "&E" by blast
1112  AOT_thus ‹τ{τ1...τn} = σ{τ1...τn}›
1113    using assms(2) "→E" by blast
1114qed
1115
1116AOT_theorem "rule-id-def:1[zero]":
1117  assumes ‹τ =df σ› and ‹σ↓›
1118  shows ‹τ = σ›
1119proof -
1120  AOT_have ‹σ↓ → τ = σ›
1121    using "df-rules-terms[4]" assms(1) "&E" by blast
1122  AOT_thus ‹τ = σ›
1123    using assms(2) "→E" by blast
1124qed
1125
1126AOT_theorem "rule-id-def:2:a":
1127  assumes ‹τ{α1...αn} =df σ{α1...αn}› and ‹σ{τ1...τn}↓› and ‹φ{τ{τ1...τn}}›
1128  shows ‹φ{σ{τ1...τn}}›
1129proof -
1130  AOT_have ‹τ{τ1...τn} = σ{τ1...τn}› using "rule-id-def:1" assms(1,2) by blast
1131  AOT_thus ‹φ{σ{τ1...τn}}› using assms(3) "rule=E" by blast
1132qed
1133
1134(* TODO: get rid of this, ideally *)
1135AOT_theorem "rule-id-def:2:a[2]":
1136  assumes ‹τ{«(α1,α2)»} =df σ{«(α1,α2)»}› and ‹σ{«(τ1,τ2)»}↓› and ‹φ{τ{«(τ1,τ2)»}}›
1137  shows ‹φ{σ{«(τ1,τ2)»}}›
1138proof -
1139  AOT_have ‹τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1140  proof -
1141    AOT_have ‹σ{«(τ1,τ2)»}↓ → τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1142      using assms by (simp add: AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def) (* NOTE: semantics needed *)
1143    AOT_thus ‹τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1144      using assms(2) "→E" by blast
1145  qed
1146  AOT_thus ‹φ{σ{«(τ1,τ2)»}}› using assms(3) "rule=E" by blast
1147qed
1148
1149AOT_theorem "rule-id-def:2:a[zero]":
1150  assumes ‹τ =df σ› and ‹σ↓› and ‹φ{τ}›
1151  shows ‹φ{σ}›
1152proof -
1153  AOT_have ‹τ = σ› using "rule-id-def:1[zero]" assms(1,2) by blast
1154  AOT_thus ‹φ{σ}› using assms(3) "rule=E" by blast
1155qed
1156
1157lemmas "=dfE" = "rule-id-def:2:a" "rule-id-def:2:a[zero]"
1158
1159AOT_theorem "rule-id-def:2:b":
1160  assumes ‹τ{α1...αn} =df σ{α1...αn}› and ‹σ{τ1...τn}↓› and ‹φ{σ{τ1...τn}}›
1161  shows ‹φ{τ{τ1...τn}}›
1162proof -
1163  AOT_have ‹τ{τ1...τn} = σ{τ1...τn}› using "rule-id-def:1" assms(1,2) by blast
1164  AOT_hence ‹σ{τ1...τn} = τ{τ1...τn}›
1165    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1166  AOT_thus ‹φ{τ{τ1...τn}}› using assms(3) "rule=E" by blast
1167qed
1168
1169(* TODO: get rid of this, ideally *)
1170AOT_theorem "rule-id-def:2:b[2]":
1171  assumes ‹τ{«(α1,α2)»} =df σ{«(α1,α2)»}› and ‹σ{«(τ1,τ2)»}↓› and ‹φ{σ{«(τ1,τ2)»}}›
1172  shows ‹φ{τ{«(τ1,τ2)»}}›
1173proof -
1174  AOT_have ‹τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1175  proof -
1176    AOT_have ‹σ{«(τ1,τ2)»}↓ → τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1177      using assms by (simp add: AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def) (* NOTE: semantics needed *)
1178    AOT_thus ‹τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1179      using assms(2) "→E" by blast
1180  qed
1181  AOT_hence ‹σ{«(τ1,τ2)»} = τ{«(τ1,τ2)»}›
1182    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1183  AOT_thus ‹φ{τ{«(τ1,τ2)»}}› using assms(3) "rule=E" by blast
1184qed
1185
1186AOT_theorem "rule-id-def:2:b[zero]":
1187  assumes ‹τ =df σ› and ‹σ↓› and ‹φ{σ}›
1188  shows ‹φ{τ}›
1189proof -
1190  AOT_have ‹τ = σ› using "rule-id-def:1[zero]" assms(1,2) by blast
1191  AOT_hence ‹σ = τ›
1192    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1193  AOT_thus ‹φ{τ}› using assms(3) "rule=E" by blast
1194qed
1195
1196lemmas "=dfI" = "rule-id-def:2:b" "rule-id-def:2:b[zero]"
1197
1198AOT_theorem "free-thms:1": ‹τ↓ ≡ ∃β (β = τ)›
1199  by (metis "∃E" "rule=I:1" "t=t-proper:2" "→I" "∃I"(1) "≡I" "→E")
1200
1201AOT_theorem "free-thms:2": ‹∀α φ{α} → (∃β (β = τ) → φ{τ})›
1202  by (metis "∃E" "rule=E" "cqt:2[const_var]"[axiom_inst] "→I" "∀E"(1))
1203
1204AOT_theorem "free-thms:3[const_var]": ‹∃β (β = α)›
1205  by (meson "∃I"(2) "id-eq:1")
1206
1207AOT_theorem "free-thms:3[lambda]": assumes ‹INSTANCE_OF_CQT_2(φ)› shows ‹∃β (β = [λν1...νn φ{ν1...νn}])›
1208  by (meson "=I"(3) assms "cqt:2[lambda]"[axiom_inst] "existential:1")
1209
1210AOT_theorem "free-thms:4[rel]": ‹([Π]κ1...κn ∨ κ1...κn[Π]) → ∃β (β = Π)›
1211  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1212
1213(* TODO: this is a rather weird way to formulate this and we don't have tuple-existential-elimination
1214         or tuple-equality-elimination in the theory... Splitting them is also a bit unfortunate, though.*)
1215AOT_theorem "free-thms:4[vars]": ‹([Π]κ1...κn ∨ κ1...κn[Π]) → ∃β1...∃βn (β1...βn = κ1...κn)›
1216  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1217
1218AOT_theorem "free-thms:4[1,rel]": ‹([Π]κ ∨ κ[Π]) → ∃β (β = Π)›
1219  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1220AOT_theorem "free-thms:4[1,1]": ‹([Π]κ ∨ κ[Π]) → ∃β (β = κ)›
1221  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1222
1223AOT_theorem "free-thms:4[2,rel]": ‹([Π]κ1κ2 ∨ κ1κ2[Π]) → ∃β (β = Π)›
1224  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1225AOT_theorem "free-thms:4[2,1]": ‹([Π]κ1κ2 ∨ κ1κ2[Π]) → ∃β (β = κ1)›
1226  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1227AOT_theorem "free-thms:4[2,2]": ‹([Π]κ1κ2 ∨ κ1κ2[Π]) → ∃β (β = κ2)›
1228  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1229AOT_theorem "free-thms:4[3,rel]": ‹([Π]κ1κ2κ3 ∨ κ1κ2κ3[Π]) → ∃β (β = Π)›
1230  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1231AOT_theorem "free-thms:4[3,1]": ‹([Π]κ1κ2κ3 ∨ κ1κ2κ3[Π]) → ∃β (β = κ1)›
1232  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1233AOT_theorem "free-thms:4[3,2]": ‹([Π]κ1κ2κ3 ∨ κ1κ2κ3[Π]) → ∃β (β = κ2)›
1234  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1235AOT_theorem "free-thms:4[3,3]": ‹([Π]κ1κ2κ3 ∨ κ1κ2κ3[Π]) → ∃β (β = κ3)›
1236  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1237AOT_theorem "free-thms:4[4,rel]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = Π)›
1238  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1239AOT_theorem "free-thms:4[4,1]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = κ1)›
1240  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1241AOT_theorem "free-thms:4[4,2]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = κ2)›
1242  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1243AOT_theorem "free-thms:4[4,3]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = κ3)›
1244  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1245AOT_theorem "free-thms:4[4,4]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = κ4)›
1246  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1247
1248AOT_theorem "ex:1:a": ‹∀α α↓›
1249  by (rule GEN) (fact "cqt:2[const_var]"[axiom_inst])
1250AOT_theorem "ex:1:b": ‹∀α∃β(β = α)›
1251  by (rule GEN) (fact "free-thms:3[const_var]")
1252
1253AOT_theorem "ex:2:a": ‹□α↓›
1254  by (rule RN) (fact "cqt:2[const_var]"[axiom_inst])
1255AOT_theorem "ex:2:b": ‹□∃β(β = α)›
1256  by (rule RN) (fact "free-thms:3[const_var]")
1257
1258AOT_theorem "ex:3:a": ‹□∀α α↓›
1259  by (rule RN) (fact "ex:1:a")
1260AOT_theorem "ex:3:b": ‹□∀α∃β(β = α)›
1261  by (rule RN) (fact "ex:1:b")
1262
1263AOT_theorem "ex:4:a": ‹∀α □α↓›
1264  by (rule GEN; rule RN) (fact "cqt:2[const_var]"[axiom_inst])
1265AOT_theorem "ex:4:b": ‹∀α□∃β(β = α)›
1266  by (rule GEN; rule RN) (fact "free-thms:3[const_var]")
1267
1268AOT_theorem "ex:5:a": ‹□∀α □α↓›
1269  by (rule RN) (simp add: "ex:4:a")
1270AOT_theorem "ex:5:b": ‹□∀α□∃β(β = α)›
1271  by (rule RN) (simp add: "ex:4:b")
1272
1273AOT_theorem "all-self=:1": ‹□∀α(α = α)›
1274  by (rule RN; rule GEN) (fact "id-eq:1")
1275AOT_theorem "all-self=:2": ‹∀α□(α = α)›
1276  by (rule GEN; rule RN) (fact "id-eq:1")
1277
1278AOT_theorem "id-nec:1": ‹α = β → □(α = β)›
1279proof(rule "→I")
1280  AOT_assume ‹α = β›
1281  moreover AOT_have ‹□(α = α)›
1282    by (rule RN) (fact "id-eq:1")
1283  ultimately AOT_show ‹□(α = β)› using "rule=E" by fast
1284qed
1285
1286AOT_theorem "id-nec:2": ‹τ = σ → □(τ = σ)›
1287proof(rule "→I")
1288  AOT_assume asm: ‹τ = σ›
1289  moreover AOT_have ‹τ↓›
1290    using calculation "t=t-proper:1" "→E" by blast
1291  moreover AOT_have ‹□(τ = τ)›
1292    using calculation "all-self=:2" "∀E"(1) by blast
1293  ultimately AOT_show ‹□(τ = σ)› using "rule=E" by fast
1294qed
1295
1296AOT_theorem "term-out:1": ‹φ{α} ≡ ∃β (β = α & φ{β})›
1297proof (rule "≡I"; rule "→I")
1298  AOT_assume asm: ‹φ{α}›
1299  AOT_show ‹∃β (β = α & φ{β})›
1300    by (rule "∃I"(2)[where β=α]; rule "&I")
1301       (auto simp: "id-eq:1" asm)
1302next
1303  AOT_assume 0: ‹∃β (β = α & φ{β})›
1304  (* TODO: have another look at this instantiation. Ideally AOT_obtain would resolve directly to the existential
1305           statement as proof obligation *)
1306  AOT_obtain β where ‹β = α & φ{β}› using "instantiation"[rotated, OF 0] by blast
1307  AOT_thus ‹φ{α}› using "&E" "rule=E" by blast
1308qed
1309
1310AOT_theorem "term-out:2": ‹τ↓ → (φ{τ} ≡ ∃α(α = τ & φ{α}))›
1311proof(rule "→I")
1312  AOT_assume ‹τ↓›
1313  moreover AOT_have ‹∀α (φ{α} ≡ ∃β (β = α & φ{β}))›
1314    by (rule GEN) (fact "term-out:1")
1315  ultimately AOT_show ‹φ{τ} ≡ ∃α(α = τ & φ{α})›
1316    using "∀E" by blast
1317qed
1318
1319(* TODO: example of an apply-style proof. Keep or reformulate? *)
1320AOT_theorem "term-out:3": ‹(φ{α} & ∀β(φ{β} → β = α)) ≡ ∀β(φ{β} ≡ β = α)›
1321  apply (rule "≡I"; rule "→I")
1322   apply (frule "&E"(1)) apply (drule "&E"(2))
1323   apply (rule GEN; rule "≡I"; rule "→I")
1324  using "rule-ui:2[const_var]" "vdash-properties:5" apply blast
1325  apply (meson "rule=E" "id-eq:1")
1326  apply (rule "&I")
1327  using "id-eq:1" "≡E"(2) "rule-ui:3" apply blast
1328  apply (rule GEN; rule "→I")
1329  using "≡E"(1) "rule-ui:2[const_var]" by blast
1330
1331AOT_theorem "term-out:4": ‹(φ{β} & ∀α(φ{α} → α = β)) ≡ ∀α(φ{α} ≡ α = β)›
1332  using "term-out:3" . (* TODO: same as above - another instance of the generalized alphabetic variant rule... *)
1333
1334(* TODO: would of course be nice to define it without the syntax magic *)
1335AOT_define AOT_exists_unique :: ‹α ⇒ φ ⇒ φ›
1336  "uniqueness:1": ‹«AOT_exists_unique φ» ≡df ∃α (φ{α} & ∀β (φ{β} → β = α))›
1337syntax "_AOT_exists_unique" :: ‹α ⇒ φ ⇒ φ› ("∃!_ _" [1,40])
1338AOT_syntax_print_translations
1339  "_AOT_exists_unique τ φ" <= "CONST AOT_exists_unique (_abs τ φ)"
1340syntax
1341   "_AOT_exists_unique_ellipse" :: ‹id_position ⇒ id_position ⇒ φ ⇒ φ› (‹∃!_...∃!_ _› [1,40])
1342parse_ast_translation‹[(syntax_const‹_AOT_exists_unique_ellipse›, fn ctx => fn [a,b,c] =>
1343  Ast.mk_appl (Ast.Constant "AOT_exists_unique") [parseEllipseList "_AOT_vars" ctx [a,b],c]),
1344(syntax_const‹_AOT_exists_unique›,AOT_restricted_binder const_name‹AOT_exists_unique› const_syntax‹AOT_conj›)]›
1345print_translation‹AOT_syntax_print_translations
1346  [AOT_preserve_binder_abs_tr' const_syntax‹AOT_exists_unique› syntax_const‹_AOT_exists_unique› (syntax_const‹_AOT_exists_unique_ellipse›, true) const_name‹AOT_conj›,
1347  AOT_binder_trans @{theory} @{binding "AOT_exists_unique_binder"} syntax_const‹_AOT_exists_unique›]
1348›
1349
1350
1351context AOT_meta_syntax
1352begin
1353notation AOT_exists_unique (binder "∃!" 20)
1354end
1355context AOT_no_meta_syntax
1356begin
1357no_notation AOT_exists_unique (binder "∃!" 20)
1358end
1359
1360AOT_theorem "uniqueness:2": ‹∃!α φ{α} ≡ ∃α∀β(φ{β} ≡ β = α)›
1361proof(rule "≡I"; rule "→I")
1362    AOT_assume ‹∃!α φ{α}›
1363    AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))›
1364      using "uniqueness:1" "≡dfE" by blast
1365    then AOT_obtain α where ‹φ{α} & ∀β (φ{β} → β = α)› using "instantiation"[rotated] by blast
1366    AOT_hence ‹∀β(φ{β} ≡ β = α)› using "term-out:3" "≡E" by blast
1367    AOT_thus ‹∃α∀β(φ{β} ≡ β = α)›
1368      using "∃I" by fast
1369next
1370    AOT_assume ‹∃α∀β(φ{β} ≡ β = α)›
1371    then AOT_obtain α where ‹∀β (φ{β} ≡ β = α)› using "instantiation"[rotated] by blast
1372    AOT_hence ‹φ{α} & ∀β (φ{β} → β = α)› using "term-out:3" "≡E" by blast
1373    AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))›
1374      using "∃I" by fast
1375    AOT_thus ‹∃!α φ{α}›
1376      using "uniqueness:1" "≡dfI" by blast
1377qed
1378
1379AOT_theorem "uni-most": ‹∃!α φ{α} → ∀β∀γ((φ{β} & φ{γ}) → β = γ)›
1380proof(rule "→I"; rule GEN; rule GEN; rule "→I")
1381  fix β γ
1382  AOT_assume ‹∃!α φ{α}›
1383  AOT_hence ‹∃α∀β(φ{β} ≡ β = α)›
1384    using "uniqueness:2" "≡E" by blast
1385  then AOT_obtain α where ‹∀β(φ{β} ≡ β = α)›
1386    using "instantiation"[rotated] by blast
1387  moreover AOT_assume ‹φ{β} & φ{γ}›
1388  ultimately AOT_have ‹β = α› and ‹γ = α›
1389    using "∀E"(2) "&E" "≡E"(1,2) by blast+
1390  AOT_thus ‹β = γ›
1391    by (metis "rule=E" "id-eq:2" "→E")
1392qed
1393
1394AOT_theorem "nec-exist-!": ‹∀α(φ{α} → □φ{α}) → (∃!α φ{α} → ∃!α □φ{α})›
1395proof (rule "→I"; rule "→I")
1396  AOT_assume a: ‹∀α(φ{α} → □φ{α})›
1397  AOT_assume ‹∃!α φ{α}›
1398  AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))› using "uniqueness:1" "≡dfE" by blast
1399  then AOT_obtain α where ξ: ‹φ{α} & ∀β (φ{β} → β = α)› using "instantiation"[rotated] by blast
1400  AOT_have ‹□φ{α}›
1401    using ξ a "&E" "∀E" "→E" by fast
1402  moreover AOT_have ‹∀β (□φ{β} → β = α)›
1403    apply (rule GEN; rule "→I")
1404    using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "→E"] "qml:2"[axiom_inst, THEN "→E"] by blast
1405  ultimately AOT_have ‹(□φ{α} & ∀β (□φ{β} → β = α))›
1406    using "&I" by blast
1407  AOT_thus ‹∃!α □φ{α}›
1408    using "uniqueness:1" "≡dfI" "∃I" by fast
1409qed
1410
1411AOT_theorem "act-cond": ‹𝒜(φ → ψ) → (𝒜φ → 𝒜ψ)›
1412  using "→I" "≡E"(1) "logic-actual-nec:2"[axiom_inst] by blast
1413
1414AOT_theorem "nec-imp-act": ‹□φ → 𝒜φ›
1415  by (metis "act-cond" "contraposition:1[2]" "≡E"(4) "qml:2"[THEN act_closure, axiom_inst] "qml-act:2"[axiom_inst] RAA(1) "→E" "→I")
1416
1417AOT_theorem "act-conj-act:1": ‹𝒜(𝒜φ → φ)›
1418  using "→I" "≡E"(2) "logic-actual-nec:2"[axiom_inst] "logic-actual-nec:4"[axiom_inst] by blast
1419
1420AOT_theorem "act-conj-act:2": ‹𝒜(φ → 𝒜φ)›
1421  by (metis "→I" "≡E"(2, 4) "logic-actual-nec:2"[axiom_inst] "logic-actual-nec:4"[axiom_inst] RAA(1))
1422
1423AOT_theorem "act-conj-act:3": ‹(𝒜φ & 𝒜ψ) → 𝒜(φ & ψ)›
1424proof -
1425  AOT_have ‹□(φ → (ψ → (φ & ψ)))›
1426    by (rule RN) (fact Adjunction)
1427  AOT_hence ‹𝒜(φ → (ψ → (φ & ψ)))›
1428    using "nec-imp-act" "→E" by blast
1429  AOT_hence ‹𝒜φ → 𝒜(ψ → (φ & ψ))›
1430    using "act-cond" "→E" by blast
1431  moreover AOT_have ‹𝒜(ψ → (φ & ψ)) → (𝒜ψ → 𝒜(φ & ψ))›
1432    by (fact "act-cond")
1433  ultimately AOT_have ‹𝒜φ → (𝒜ψ → 𝒜(φ & ψ))›
1434    using "→I" "→E" by metis
1435  AOT_thus ‹(𝒜φ & 𝒜ψ) → 𝒜(φ & ψ)›
1436    by (metis Importation "→E")
1437qed
1438
1439AOT_theorem "act-conj-act:4": ‹𝒜(𝒜φ ≡ φ)›
1440proof -
1441  AOT_have ‹(𝒜(𝒜φ → φ) & 𝒜(φ → 𝒜φ)) → 𝒜((𝒜φ → φ) & (φ → 𝒜φ))›
1442    by (fact "act-conj-act:3")
1443  moreover AOT_have ‹𝒜(𝒜φ → φ) & 𝒜(φ → 𝒜φ)›
1444    using "&I" "act-conj-act:1" "act-conj-act:2" by simp
1445  ultimately AOT_have ζ: ‹𝒜((𝒜φ → φ) & (φ → 𝒜φ))›
1446    using "→E" by blast
1447  AOT_have ‹𝒜(((𝒜φ → φ) & (φ → 𝒜φ)) → (𝒜φ ≡ φ))›
1448    using "conventions:3"[THEN "df-rules-formulas[2]", THEN act_closure, axiom_inst] by blast
1449  AOT_hence ‹𝒜((𝒜φ → φ) & (φ → 𝒜φ)) → 𝒜(𝒜φ ≡ φ)›
1450    using "act-cond" "→E" by blast
1451  AOT_thus ‹𝒜(𝒜φ ≡ φ)› using ζ "→E" by blast
1452qed
1453
1454(* TODO: consider introducing AOT_inductive *)
1455inductive arbitrary_actualization for φ where
1456  ‹arbitrary_actualization φ «𝒜φ»›
1457| ‹arbitrary_actualization φ «𝒜ψ»› if ‹arbitrary_actualization φ ψ›
1458declare arbitrary_actualization.cases[AOT] arbitrary_actualization.induct[AOT]
1459        arbitrary_actualization.simps[AOT] arbitrary_actualization.intros[AOT]
1460syntax arbitrary_actualization :: ‹φ' ⇒ φ' ⇒ AOT_prop› ("ARBITRARY'_ACTUALIZATION'(_,_')")
1461
1462notepad
1463begin
1464  AOT_modally_strict {
1465    fix φ
1466    AOT_have ‹ARBITRARY_ACTUALIZATION(𝒜φ ≡ φ, 𝒜(𝒜φ ≡ φ))›
1467      using AOT_PLM.arbitrary_actualization.intros by metis
1468    AOT_have ‹ARBITRARY_ACTUALIZATION(𝒜φ ≡ φ, 𝒜𝒜(𝒜φ ≡ φ))›
1469      using AOT_PLM.arbitrary_actualization.intros by metis
1470    AOT_have ‹ARBITRARY_ACTUALIZATION(𝒜φ ≡ φ, 𝒜𝒜𝒜(𝒜φ ≡ φ))›
1471      using AOT_PLM.arbitrary_actualization.intros by metis
1472  }
1473end
1474
1475
1476AOT_theorem "closure-act:1": assumes ‹ARBITRARY_ACTUALIZATION(𝒜φ ≡ φ, ψ)› shows ‹ψ›
1477using assms proof(induct)
1478  case 1
1479  AOT_show ‹𝒜(𝒜φ ≡ φ)›
1480    by (simp add: "act-conj-act:4")
1481next
1482  case (2 ψ)
1483  AOT_thus ‹𝒜ψ›
1484    by (metis arbitrary_actualization.simps "≡E"(1) "logic-actual-nec:4"[axiom_inst])
1485qed
1486
1487AOT_theorem "closure-act:2": ‹∀α 𝒜(𝒜φ{α} ≡ φ{α})›
1488  by (simp add: "act-conj-act:4" "∀I")
1489
1490AOT_theorem "closure-act:3": ‹𝒜∀α 𝒜(𝒜φ{α} ≡ φ{α})›
1491  by (metis (no_types, lifting) "act-conj-act:4" "≡E"(1,2) "logic-actual-nec:3"[axiom_inst] "logic-actual-nec:4"[axiom_inst] "∀I")
1492
1493AOT_theorem "closure-act:4": ‹𝒜∀α1...∀αn 𝒜(𝒜φ{α1...αn} ≡ φ{α1...αn})›
1494  using "closure-act:3" .
1495
1496(* TODO: examine these proofs *)
1497AOT_theorem "RA[1]": assumes ‹⊢ φ› shows ‹⊢ 𝒜φ›
1498  (* This proof is the one rejected in remark (136) (meta-rule) *)
1499  using "¬¬E" assms "≡E"(3) "logic-actual"[act_axiom_inst] "logic-actual-nec:1"[axiom_inst] "modus-tollens:2" by blast
1500AOT_theorem "RA[2]": assumes ‹⊢□ φ› shows ‹𝒜φ›
1501  (* This is actually Γ ⊢□ φ ⟹ □Γ ⊢□ 𝒜φ*)
1502  using RN assms "nec-imp-act" "vdash-properties:5" by blast
1503AOT_theorem "RA[3]": assumes ‹Γ ⊢□ φ› shows ‹𝒜Γ ⊢□ 𝒜φ›
1504  using assms by (meson AOT_sem_act imageI)
1505  (* This is not exactly right either. *)
1506
1507AOT_act_theorem "ANeg:1": ‹¬𝒜φ ≡ ¬φ›
1508  by (simp add: "RA[1]" "contraposition:1[1]" "deduction-theorem" "≡I" "logic-actual"[act_axiom_inst])
1509
1510AOT_act_theorem "ANeg:2": ‹¬𝒜¬φ ≡ φ›
1511  using "ANeg:1" "≡I" "≡E"(5) "useful-tautologies:1" "useful-tautologies:2" by blast
1512
1513AOT_theorem "Act-Basic:1": ‹𝒜φ ∨ 𝒜¬φ›
1514  by (meson "∨I"(1,2) "≡E"(2) "logic-actual-nec:1"[axiom_inst] "raa-cor:1")
1515
1516AOT_theorem "Act-Basic:2": ‹𝒜(φ & ψ) ≡ (𝒜φ & 𝒜ψ)›
1517proof (rule "≡I"; rule "→I")
1518  AOT_assume ‹𝒜(φ & ψ)›
1519  moreover AOT_have ‹𝒜((φ & ψ) → φ)›
1520    by (simp add: "RA[2]" "Conjunction Simplification"(1))
1521  moreover AOT_have ‹𝒜((φ & ψ) → ψ)›
1522    by (simp add: "RA[2]" "Conjunction Simplification"(2))
1523  ultimately AOT_show ‹𝒜φ & 𝒜ψ›
1524    using "act-cond"[THEN "→E", THEN "→E"] "&I" by metis
1525next
1526  AOT_assume ‹𝒜φ & 𝒜ψ›
1527  AOT_thus ‹𝒜(φ & ψ)›
1528    using "act-conj-act:3" "vdash-properties:6" by blast
1529qed
1530
1531AOT_theorem "Act-Basic:3": ‹𝒜(φ ≡ ψ) ≡ (𝒜(φ → ψ) & 𝒜(ψ → φ))›
1532proof (rule "≡I"; rule "→I")
1533  AOT_assume ‹𝒜(φ ≡ ψ)›
1534  moreover AOT_have ‹𝒜((φ ≡ ψ) → (φ → ψ))›
1535    by (simp add: "RA[2]" "deduction-theorem" "≡E"(1))
1536  moreover AOT_have ‹𝒜((φ ≡ ψ) → (ψ → φ))›
1537    by (simp add: "RA[2]" "deduction-theorem" "≡E"(2))
1538  ultimately AOT_show ‹𝒜(φ → ψ) & 𝒜(ψ → φ)›
1539    using "act-cond"[THEN "→E", THEN "→E"] "&I" by metis
1540next
1541  AOT_assume ‹𝒜(φ → ψ) & 𝒜(ψ → φ)›
1542  AOT_hence ‹𝒜((φ → ψ) & (ψ → φ))›
1543    by (metis "act-conj-act:3" "vdash-properties:10")
1544  moreover AOT_have ‹𝒜(((φ → ψ) & (ψ → φ)) → (φ ≡ ψ))›
1545    by (simp add: "conventions:3" "RA[2]" "df-rules-formulas[2]" "vdash-properties:1[2]")
1546  ultimately AOT_show ‹𝒜(φ ≡ ψ)›
1547    using "act-cond"[THEN "→E", THEN "→E"] by metis
1548qed
1549
1550AOT_theorem "Act-Basic:4": ‹(𝒜(φ → ψ) & 𝒜(ψ → φ)) ≡ (𝒜φ ≡ 𝒜ψ)›
1551proof (rule "≡I"; rule "→I")
1552  AOT_assume 0: ‹𝒜(φ → ψ) & 𝒜(ψ → φ)›
1553  AOT_show ‹𝒜φ ≡ 𝒜ψ›
1554    using 0 "&E" "act-cond"[THEN "→E", THEN "→E"] "≡I" "→I" by metis
1555next
1556  AOT_assume ‹𝒜φ ≡ 𝒜ψ›
1557  AOT_thus ‹𝒜(φ → ψ) & 𝒜(ψ → φ)›
1558    by (metis "→I" "logic-actual-nec:2"[axiom_inst] "≡E"(1,2) "&I")
1559qed
1560
1561AOT_theorem "Act-Basic:5": ‹𝒜(φ ≡ ψ) ≡ (𝒜φ ≡ 𝒜ψ)›
1562  using "Act-Basic:3" "Act-Basic:4" "≡E"(5) by blast
1563
1564AOT_theorem "Act-Basic:6": ‹𝒜φ ≡ □𝒜φ›
1565  by (simp add: "≡I" "qml:2"[axiom_inst] "qml-act:1"[axiom_inst])
1566
1567AOT_theorem "Act-Basic:7": ‹𝒜□φ → □𝒜φ›
1568  by (metis "Act-Basic:6" "→I" "→E" "≡E"(1,2) "nec-imp-act" "qml-act:2"[axiom_inst])
1569
1570AOT_theorem "Act-Basic:8": ‹□φ → □𝒜φ›
1571  using "Hypothetical Syllogism" "nec-imp-act" "qml-act:1"[axiom_inst] by blast
1572
1573AOT_theorem "Act-Basic:9": ‹𝒜(φ ∨ ψ) ≡ (𝒜φ ∨ 𝒜ψ)›
1574proof (rule "≡I"; rule "→I")
1575  AOT_assume ‹𝒜(φ ∨ ψ)›
1576  AOT_thus ‹𝒜φ ∨ 𝒜ψ›
1577  proof (rule "raa-cor:3")
1578    AOT_assume ‹¬(𝒜φ ∨ 𝒜ψ)›
1579    AOT_hence ‹¬𝒜φ & ¬𝒜ψ›
1580      by (metis "≡E"(1) "oth-class-taut:5:d")
1581    AOT_hence ‹𝒜¬φ & 𝒜¬ψ›
1582      using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] "&E" "&I" by metis
1583    AOT_hence ‹𝒜(¬φ & ¬ψ)›
1584      using "≡E" "Act-Basic:2" by metis
1585    moreover AOT_have ‹𝒜((¬φ & ¬ψ) ≡ ¬(φ ∨ ψ))›
1586      using "RA[2]" "≡E"(6) "oth-class-taut:3:a" "oth-class-taut:5:d" by blast
1587    moreover AOT_have ‹𝒜(¬φ & ¬ψ) ≡ 𝒜(¬(φ ∨ ψ))›
1588      using calculation(2) by (metis "Act-Basic:5" "≡E"(1))
1589    ultimately AOT_have ‹𝒜(¬(φ ∨ ψ))› using "≡E" by blast
1590    AOT_thus ‹¬𝒜(φ ∨ ψ)›
1591      using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(1)] by auto
1592  qed
1593next
1594  AOT_assume ‹𝒜φ ∨ 𝒜ψ›
1595  AOT_thus ‹𝒜(φ ∨ ψ)›
1596    by (meson "RA[2]" "act-cond" "∨I"(1) "∨E"(1) "Disjunction Addition"(1) "Disjunction Addition"(2))
1597qed
1598
1599AOT_theorem "Act-Basic:10": ‹𝒜∃α φ{α} ≡ ∃α 𝒜φ{α}›
1600proof -
1601  AOT_have θ: ‹¬𝒜∀α ¬φ{α} ≡ ¬∀α 𝒜¬φ{α}›
1602    by (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
1603       (metis "logic-actual-nec:3"[axiom_inst])
1604  AOT_have ξ: ‹¬∀α 𝒜¬φ{α} ≡ ¬∀α ¬𝒜φ{α}›
1605    by (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
1606       (rule "logic-actual-nec:1"[THEN universal_closure, axiom_inst, THEN "cqt-basic:3"[THEN "→E"]])
1607  AOT_have ‹𝒜(∃α φ{α}) ≡ 𝒜(¬∀α ¬φ{α})›
1608    using "conventions:4"[THEN "df-rules-formulas[1]", THEN act_closure, axiom_inst]
1609          "conventions:4"[THEN "df-rules-formulas[2]", THEN act_closure, axiom_inst]
1610    "Act-Basic:4"[THEN "≡E"(1)] "&I" "Act-Basic:5"[THEN "≡E"(2)] by metis
1611  also AOT_have ‹… ≡ ¬𝒜∀α ¬φ{α}›
1612    by (simp add: "logic-actual-nec:1" "vdash-properties:1[2]")
1613  also AOT_have ‹… ≡ ¬∀α 𝒜 ¬φ{α}› using θ by blast
1614  also AOT_have ‹… ≡ ¬∀α ¬𝒜 φ{α}› using ξ by blast
1615  also AOT_have ‹… ≡ ∃α 𝒜 φ{α}›
1616    using "conventions:4"[THEN "≡Df"] by (metis "≡E"(6) "oth-class-taut:3:a")
1617  finally AOT_show ‹𝒜∃α φ{α} ≡ ∃α 𝒜φ{α}› .
1618qed
1619
1620
1621AOT_theorem "Act-Basic:11": ‹𝒜∀α(φ{α} ≡ ψ{α}) ≡ ∀α(𝒜φ{α} ≡ 𝒜ψ{α})›
1622proof(rule "≡I"; rule "→I")
1623  AOT_assume ‹𝒜∀α(φ{α} ≡ ψ{α})›
1624  AOT_hence ‹∀α𝒜(φ{α} ≡ ψ{α})›
1625    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(1)] by blast
1626  AOT_hence ‹𝒜(φ{α} ≡ ψ{α})› for α using "∀E" by blast
1627  AOT_hence ‹𝒜φ{α} ≡ 𝒜ψ{α}› for α by (metis "Act-Basic:5" "≡E"(1))
1628  AOT_thus ‹∀α(𝒜φ{α} ≡ 𝒜ψ{α})› by (rule "∀I")
1629next
1630  AOT_assume ‹∀α(𝒜φ{α} ≡ 𝒜ψ{α})›
1631  AOT_hence ‹𝒜φ{α} ≡ 𝒜ψ{α}› for α using "∀E" by blast
1632  AOT_hence ‹𝒜(φ{α} ≡ ψ{α})› for α by (metis "Act-Basic:5" "≡E"(2))
1633  AOT_hence ‹∀α 𝒜(φ{α} ≡ ψ{α})› by (rule "∀I")
1634  AOT_thus ‹𝒜∀α(φ{α} ≡ ψ{α})›
1635    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(2)] by fast
1636qed
1637
1638AOT_act_theorem "act-quant-uniq": ‹∀β(𝒜φ{β} ≡ β = α) ≡ ∀β(φ{β} ≡ β = α)›
1639proof(rule "≡I"; rule "→I")
1640  AOT_assume ‹∀β(𝒜φ{β} ≡ β = α)›
1641  AOT_hence ‹𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1642  AOT_hence ‹φ{β} ≡ β = α› for β
1643    using "≡I" "→I" "RA[1]" "≡E"(1) "≡E"(2) "logic-actual"[act_axiom_inst] "vdash-properties:6"
1644    by metis
1645  AOT_thus ‹∀β(φ{β} ≡ β = α)› by (rule "∀I")
1646next
1647  AOT_assume ‹∀β(φ{β} ≡ β = α)›
1648  AOT_hence ‹φ{β} ≡ β = α› for β using "∀E" by blast
1649  AOT_hence ‹𝒜φ{β} ≡ β = α› for β
1650    using "≡I" "→I" "RA[1]" "≡E"(1) "≡E"(2) "logic-actual"[act_axiom_inst] "vdash-properties:6"
1651    by metis
1652  AOT_thus ‹∀β(𝒜φ{β} ≡ β = α)› by (rule "∀I")
1653qed
1654
1655AOT_act_theorem "fund-cont-desc": ‹x = ιx(φ{x}) ≡ ∀z(φ{z} ≡ z = x)›
1656  using descriptions[axiom_inst] "act-quant-uniq" "≡E"(5) by fast
1657
1658AOT_act_theorem hintikka: ‹x = ιx(φ{x}) ≡ (φ{x} & ∀z (φ{z} → z = x))›
1659  using "Commutativity of ≡"[THEN "≡E"(1)] "term-out:3" "fund-cont-desc" "≡E"(5) by blast
1660
1661
1662locale russel_axiom =
1663  fixes ψ
1664  assumes ψ_denotes_asm: "[v ⊨ ψ{κ}] ⟹ [v ⊨ κ↓]"
1665begin
1666AOT_act_theorem "russell-axiom": ‹ψ{ιx φ{x}} ≡ ∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})›
1667proof -
1668  AOT_have b: ‹∀x (x = ιx φ{x} ≡ (φ{x} & ∀z(φ{z} → z = x)))›
1669    using hintikka "∀I" by fast
1670  show ?thesis
1671  proof(rule "≡I"; rule "→I")
1672    AOT_assume c: ‹ψ{ιx φ{x}}›
1673    AOT_hence d: ‹ιx φ{x}↓› using ψ_denotes_asm by blast
1674    AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1675    then AOT_obtain a where a_def: ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1676    moreover AOT_have ‹a = ιx φ{x} ≡ (φ{a} & ∀z(φ{z} → z = a))› using b "∀E" by blast
1677    ultimately AOT_have ‹φ{a} & ∀z(φ{z} → z = a)› using "≡E" by blast
1678    moreover AOT_have ‹ψ{a}›
1679    proof - 
1680      AOT_have 1: ‹∀x∀y(x = y → y = x)›
1681        by (simp add: "id-eq:2" "universal-cor")
1682      AOT_have ‹a = ιx φ{x} →  ιx φ{x} = a›
1683        by (rule "∀E"(1)[where τ="«ιx φ{x}»"]; rule "∀E"(2)[where β=a])
1684           (auto simp: 1 d "universal-cor")
1685      AOT_thus ‹ψ{a}›
1686        using a_def c "rule=E" "→E" by blast
1687    qed
1688    ultimately AOT_have ‹φ{a} & ∀z(φ{z} → z = a) & ψ{a}› by (rule "&I")
1689    AOT_thus ‹∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})› by (rule "∃I")
1690  next
1691    AOT_assume ‹∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})›
1692    then AOT_obtain b where g: ‹φ{b} & ∀z(φ{z} → z = b) & ψ{b}› using "instantiation"[rotated] by blast
1693    AOT_hence h: ‹b = ιx φ{x} ≡ (φ{b} & ∀z(φ{z} → z = b))› using b "∀E" by blast
1694    AOT_have ‹φ{b} & ∀z(φ{z} → z = b)› and j: ‹ψ{b}› using g "&E" by blast+
1695    AOT_hence ‹b = ιx φ{x}› using h "≡E" by blast
1696    AOT_thus ‹ψ{ιx φ{x}}› using j "rule=E" by blast
1697  qed
1698qed
1699end
1700
1701(* TODO: this nicely shows off using locales with the embedding, but maybe there is still a nicer way *)
1702(* TODO: sledgehammer tends to refer to ψ_denotes_asm in these instantiation instead of referring
1703         to cqt:5:a - should be fixed *)
1704interpretation "russell-axiom[exe,1]": russel_axiom ‹λ κ . «[Π]κ»›
1705  by standard (metis "cqt:5:a[1]"[axiom_inst, THEN "→E"] "&E"(2))
1706interpretation "russell-axiom[exe,2,1,1]": russel_axiom ‹λ κ . «[Π]κκ'»›
1707  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E")
1708interpretation "russell-axiom[exe,2,1,2]": russel_axiom ‹λ κ . «[Π]κ'κ»›
1709  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E"(2))
1710interpretation "russell-axiom[exe,2,2]": russel_axiom ‹λ κ . «[Π]κκ»›
1711  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E"(2))
1712interpretation "russell-axiom[exe,3,1,1]": russel_axiom ‹λ κ . «[Π]κκ'κ''»›
1713  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1714interpretation "russell-axiom[exe,3,1,2]": russel_axiom ‹λ κ . «[Π]κ'κκ''»›
1715  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1716interpretation "russell-axiom[exe,3,1,3]": russel_axiom ‹λ κ . «[Π]κ'κ''κ»›
1717  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1718interpretation "russell-axiom[exe,3,2,1]": russel_axiom ‹λ κ . «[Π]κκκ'»›
1719  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1720interpretation "russell-axiom[exe,3,2,2]": russel_axiom ‹λ κ . «[Π]κκ'κ»›
1721  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1722interpretation "russell-axiom[exe,3,2,3]": russel_axiom ‹λ κ . «[Π]κ'κκ»›
1723  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1724interpretation "russell-axiom[exe,3,3]": russel_axiom ‹λ κ . «[Π]κκκ»›
1725  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1726
1727interpretation "russell-axiom[enc,1]": russel_axiom ‹λ κ . «κ[Π]»›
1728  by standard (metis "cqt:5:b[1]"[axiom_inst, THEN "→E"] "&E"(2))
1729interpretation "russell-axiom[enc,2,1]": russel_axiom ‹λ κ . «κκ'[Π]»›
1730  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E")
1731interpretation "russell-axiom[enc,2,2]": russel_axiom ‹λ κ . «κ'κ[Π]»›
1732  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E"(2))
1733interpretation "russell-axiom[enc,2,3]": russel_axiom ‹λ κ . «κκ[Π]»›
1734  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E"(2))
1735interpretation "russell-axiom[enc,3,1,1]": russel_axiom ‹λ κ . «κκ'κ''[Π]»›
1736  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1737interpretation "russell-axiom[enc,3,1,2]": russel_axiom ‹λ κ . «κ'κκ''[Π]»›
1738  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1739interpretation "russell-axiom[enc,3,1,3]": russel_axiom ‹λ κ . «κ'κ''κ[Π]»›
1740  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1741interpretation "russell-axiom[enc,3,2,1]": russel_axiom ‹λ κ . «κκκ'[Π]»›
1742  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1743interpretation "russell-axiom[enc,3,2,2]": russel_axiom ‹λ κ . «κκ'κ[Π]»›
1744  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1745interpretation "russell-axiom[enc,3,2,3]": russel_axiom ‹λ κ . «κ'κκ[Π]»›
1746  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1747interpretation "russell-axiom[enc,3,3]": russel_axiom ‹λ κ . «κκκ[Π]»›
1748  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1749
1750AOT_act_theorem "1-exists:1": ‹ιx φ{x}↓ ≡ ∃!x φ{x}›
1751proof(rule "≡I"; rule "→I")
1752  AOT_assume ‹ιx φ{x}↓›
1753  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1754  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1755  AOT_hence ‹φ{a} & ∀z (φ{z} → z = a)› using hintikka "≡E" by blast
1756  AOT_hence ‹∃x (φ{x} & ∀z (φ{z} → z = x))› by (rule "∃I")
1757  AOT_thus ‹∃!x φ{x}› using "uniqueness:1"[THEN "≡dfI"] by blast
1758next
1759  AOT_assume ‹∃!x φ{x}›
1760  AOT_hence ‹∃x (φ{x} & ∀z (φ{z} → z = x))›
1761    using "uniqueness:1"[THEN "≡dfE"] by blast
1762  then AOT_obtain b where ‹φ{b} & ∀z (φ{z} → z = b)› using "instantiation"[rotated] by blast
1763  AOT_hence ‹b = ιx φ{x}› using hintikka "≡E" by blast
1764  AOT_thus ‹ιx φ{x}↓› by (metis "t=t-proper:2" "vdash-properties:6")
1765qed
1766
1767AOT_act_theorem "1-exists:2": ‹∃y(y=ιx φ{x}) ≡ ∃!x φ{x}›
1768  using "1-exists:1" "free-thms:1" "≡E"(6) by blast
1769
1770AOT_act_theorem "y-in:1": ‹x = ιx φ{x} → φ{x}›
1771  using "&E"(1) "→I" hintikka "≡E"(1) by blast
1772
1773AOT_act_theorem "y-in:2": ‹z = ιx φ{x} → φ{z}› using "y-in:1". (* TODO: same as above *)
1774
1775AOT_act_theorem "y-in:3": ‹ιx φ{x}↓ → φ{ιx φ{x}}›
1776proof(rule "→I")
1777  AOT_assume ‹ιx φ{x}↓›
1778  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1779  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1780  moreover AOT_have ‹φ{a}› using calculation hintikka "≡E"(1) "&E" by blast
1781  ultimately AOT_show ‹φ{ιx φ{x}}› using "rule=E" by blast
1782qed
1783
1784AOT_act_theorem "y-in:4": ‹∃y (y = ιx φ{x}) → φ{ιx φ{x}}›
1785  using "y-in:3"[THEN "→E"] "free-thms:1"[THEN "≡E"(2)] "→I" by blast
1786
1787
1788AOT_theorem "act-quant-nec": ‹∀β (𝒜φ{β} ≡ β = α) ≡ ∀β(𝒜𝒜φ{β} ≡ β = α)›
1789proof(rule "≡I"; rule "→I")
1790  AOT_assume ‹∀β (𝒜φ{β} ≡ β = α)›
1791  AOT_hence ‹𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1792  AOT_hence ‹𝒜𝒜φ{β} ≡ β = α› for β 
1793    by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(5))
1794  AOT_thus ‹∀β(𝒜𝒜φ{β} ≡ β = α)›
1795    by (rule "∀I")
1796next
1797  AOT_assume ‹∀β(𝒜𝒜φ{β} ≡ β = α)›
1798  AOT_hence ‹𝒜𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1799  AOT_hence ‹𝒜φ{β} ≡ β = α› for β
1800    by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(6))
1801  AOT_thus ‹∀β (𝒜φ{β} ≡ β = α)›
1802    by (rule "∀I")
1803qed
1804
1805AOT_theorem "equi-desc-descA:1": ‹x = ιx φ{x} ≡ x = ιx(𝒜φ{x})›
1806proof -
1807  AOT_have ‹x = ιx φ{x} ≡ ∀z (𝒜φ{z} ≡ z = x)›  using descriptions[axiom_inst] by blast
1808  also AOT_have ‹... ≡ ∀z (𝒜𝒜φ{z} ≡ z = x)›
1809  proof(rule "≡I"; rule "→I"; rule "∀I")
1810    AOT_assume ‹∀z (𝒜φ{z} ≡ z = x)›
1811    AOT_hence ‹𝒜φ{a} ≡ a = x› for a using "∀E" by blast
1812    AOT_thus ‹𝒜𝒜φ{a} ≡ a = x› for a by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(5))
1813  next
1814    AOT_assume ‹∀z (𝒜𝒜φ{z} ≡ z = x)›
1815    AOT_hence ‹𝒜𝒜φ{a} ≡ a = x› for a using "∀E" by blast
1816    AOT_thus ‹𝒜φ{a} ≡ a = x› for a  by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(6))
1817  qed
1818  also AOT_have ‹... ≡ x = ιx(𝒜φ{x})›
1819    using "Commutativity of ≡"[THEN "≡E"(1)] descriptions[axiom_inst] by fast
1820  finally show ?thesis .
1821qed
1822
1823AOT_theorem "equi-desc-descA:2": ‹ιx φ{x}↓ → ιx φ{x} = ιx(𝒜φ{x})›
1824proof(rule "→I")
1825  AOT_assume ‹ιx φ{x}↓›
1826  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1827  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1828  moreover AOT_have ‹a = ιx(𝒜φ{x})› using calculation "equi-desc-descA:1"[THEN "≡E"(1)] by blast
1829  ultimately AOT_show ‹ιx φ{x} = ιx(𝒜φ{x})› using "rule=E" by fast
1830qed
1831
1832AOT_theorem "nec-hintikka-scheme": ‹x = ιx φ{x} ≡ 𝒜φ{x} & ∀z(𝒜φ{z} → z = x)›
1833proof -
1834  AOT_have ‹x = ιx φ{x} ≡ ∀z(𝒜φ{z} ≡ z = x)› using descriptions[axiom_inst] by blast
1835  also AOT_have ‹… ≡ (𝒜φ{x} & ∀z(𝒜φ{z} → z = x))›
1836    using "Commutativity of ≡"[THEN "≡E"(1)] "term-out:3" by fast
1837  finally show ?thesis.
1838qed
1839
1840AOT_theorem "equiv-desc-eq:1": ‹𝒜∀x(φ{x} ≡ ψ{x}) → ∀x (x = ιx φ{x} ≡ x = ιx ψ{x})›
1841proof(rule "→I"; rule "∀I")
1842  fix β
1843  AOT_assume ‹𝒜∀x(φ{x} ≡ ψ{x})›
1844  AOT_hence ‹𝒜(φ{x} ≡ ψ{x})› for x using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(1)] "∀E"(2) by blast
1845  AOT_hence 0: ‹𝒜φ{x} ≡ 𝒜ψ{x}› for x by (metis "Act-Basic:5" "≡E"(1))
1846  AOT_have ‹β = ιx φ{x} ≡ 𝒜φ{β} & ∀z(𝒜φ{z} → z = β)› using "nec-hintikka-scheme" by blast
1847  also AOT_have ‹... ≡ 𝒜ψ{β} & ∀z(𝒜ψ{z} → z = β)›
1848  proof (rule "≡I"; rule "→I")
1849    AOT_assume 1: ‹𝒜φ{β} & ∀z(𝒜φ{z} → z = β)›
1850    AOT_hence ‹𝒜φ{z} → z = β› for z using "&E" "∀E" by blast
1851    AOT_hence ‹𝒜ψ{z} → z = β› for z using 0 "≡E" "→I" "→E" by metis
1852    AOT_hence ‹∀z(𝒜ψ{z} → z = β)› using "∀I" by fast
1853    moreover AOT_have ‹𝒜ψ{β}› using "&E" 0[THEN "≡E"(1)] 1 by blast
1854    ultimately AOT_show ‹𝒜ψ{β} & ∀z(𝒜ψ{z} → z = β)› using "&I" by blast
1855  next
1856    AOT_assume 1: ‹𝒜ψ{β} & ∀z(𝒜ψ{z} → z = β)›
1857    AOT_hence ‹𝒜ψ{z} → z = β› for z using "&E" "∀E" by blast
1858    AOT_hence ‹𝒜φ{z} → z = β› for z using 0 "≡E" "→I" "→E" by metis
1859    AOT_hence ‹∀z(𝒜φ{z} → z = β)› using "∀I" by fast
1860    moreover AOT_have ‹𝒜φ{β}› using "&E" 0[THEN "≡E"(2)] 1 by blast
1861    ultimately AOT_show ‹𝒜φ{β} & ∀z(𝒜φ{z} → z = β)› using "&I" by blast
1862  qed
1863  also AOT_have ‹... ≡ β = ιx ψ{x}›
1864    using "Commutativity of ≡"[THEN "≡E"(1)] "nec-hintikka-scheme" by blast
1865  finally AOT_show ‹β = ιx φ{x} ≡ β = ιx ψ{x}› .
1866qed
1867
1868AOT_theorem "equiv-desc-eq:2": ‹ιx φ{x}↓ & 𝒜∀x(φ{x} ≡ ψ{x}) → ιx φ{x} = ιx ψ{x}›
1869proof(rule "→I")
1870  AOT_assume ‹ιx φ{x}↓ & 𝒜∀x(φ{x} ≡ ψ{x})›
1871  AOT_hence 0: ‹∃y (y = ιx φ{x})› and
1872            1: ‹∀x (x = ιx φ{x} ≡ x = ιx ψ{x})›
1873    using "&E" "free-thms:1"[THEN "≡E"(1)] "equiv-desc-eq:1" "→E" by blast+
1874  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1875  moreover AOT_have ‹a = ιx ψ{x}› using calculation 1 "∀E" "≡E"(1) by fast
1876  ultimately AOT_show ‹ιx φ{x} = ιx ψ{x}›
1877    using "rule=E" by fast
1878qed
1879
1880AOT_theorem "equiv-desc-eq:3": ‹ιx φ{x}↓ & □∀x(φ{x} ≡ ψ{x}) → ιx φ{x} = ιx ψ{x}›
1881  using "→I" "equiv-desc-eq:2"[THEN "→E", OF "&I"] "&E" "nec-imp-act"[THEN "→E"] by metis
1882
1883(* Note: this is a special case of "exist-nec" *)
1884AOT_theorem "equiv-desc-eq:4": ‹ιx φ{x}↓ → □ιx φ{x}↓›
1885proof(rule "→I")
1886  AOT_assume ‹ιx φ{x}↓›
1887  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1888  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1889  AOT_thus ‹□ιx φ{x}↓›
1890    using "ex:2:a" "rule=E" by fast
1891qed
1892
1893AOT_theorem "equiv-desc-eq:5": ‹ιx φ{x}↓ → ∃y □(y = ιx φ{x})›
1894proof(rule "→I")
1895  AOT_assume ‹ιx φ{x}↓›
1896  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1897  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1898  AOT_hence ‹□(a = ιx φ{x})› by (metis "id-nec:2" "vdash-properties:10")
1899  AOT_thus ‹∃y □(y = ιx φ{x})› by (rule "∃I")
1900qed
1901
1902AOT_act_theorem "equiv-desc-eq2:1": ‹∀x (φ{x} ≡ ψ{x}) → ∀x (x = ιx φ{x} ≡ x = ιx ψ{x})›
1903  using "→I" "logic-actual"[act_axiom_inst, THEN "→E"] "equiv-desc-eq:1"[THEN "→E"]
1904        "RA[1]" "deduction-theorem" by blast
1905
1906AOT_act_theorem "equiv-desc-eq2:2": ‹ιx φ{x}↓ & ∀x (φ{x} ≡ ψ{x}) → ιx φ{x} = ιx ψ{x}›
1907  using "→I" "logic-actual"[act_axiom_inst, THEN "→E"] "equiv-desc-eq:2"[THEN "→E", OF "&I"]
1908        "RA[1]" "deduction-theorem" "&E" by metis
1909
1910context russel_axiom
1911begin
1912AOT_theorem "nec-russell-axiom": ‹ψ{ιx φ{x}} ≡ ∃x(𝒜φ{x} & ∀z(𝒜φ{z} → z = x) & ψ{x})›
1913proof -
1914  AOT_have b: ‹∀x (x = ιx φ{x} ≡ (𝒜φ{x} & ∀z(𝒜φ{z} → z = x)))›
1915    using "nec-hintikka-scheme" "∀I" by fast
1916  show ?thesis
1917  proof(rule "≡I"; rule "→I")
1918    AOT_assume c: ‹ψ{ιx φ{x}}›
1919    AOT_hence d: ‹ιx φ{x}↓› using ψ_denotes_asm by blast
1920    AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1921    then AOT_obtain a where a_def: ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1922    moreover AOT_have ‹a = ιx φ{x} ≡ (𝒜φ{a} & ∀z(𝒜φ{z} → z = a))› using b "∀E" by blast
1923    ultimately AOT_have ‹𝒜φ{a} & ∀z(𝒜φ{z} → z = a)› using "≡E" by blast
1924    moreover AOT_have ‹ψ{a}›
1925    proof - 
1926      AOT_have 1: ‹∀x∀y(x = y → y = x)›
1927        by (simp add: "id-eq:2" "universal-cor")
1928      AOT_have ‹a = ιx φ{x} →  ιx φ{x} = a›
1929        by (rule "∀E"(1)[where τ="«ιx φ{x}»"]; rule "∀E"(2)[where β=a])
1930           (auto simp: d "universal-cor" 1)
1931      AOT_thus ‹ψ{a}›
1932        using a_def c "rule=E" "→E" by metis
1933    qed
1934    ultimately AOT_have ‹𝒜φ{a} & ∀z(𝒜φ{z} → z = a) & ψ{a}› by (rule "&I")
1935    AOT_thus ‹∃x(𝒜φ{x} & ∀z(𝒜φ{z} → z = x) & ψ{x})› by (rule "∃I")
1936  next
1937    AOT_assume ‹∃x(𝒜φ{x} & ∀z(𝒜φ{z} → z = x) & ψ{x})›
1938    then AOT_obtain b where g: ‹𝒜φ{b} & ∀z(𝒜φ{z} → z = b) & ψ{b}› using "instantiation"[rotated] by blast
1939    AOT_hence h: ‹b = ιx φ{x} ≡ (𝒜φ{b} & ∀z(𝒜φ{z} → z = b))› using b "∀E" by blast
1940    AOT_have ‹𝒜φ{b} & ∀z(𝒜φ{z} → z = b)› and j: ‹ψ{b}› using g "&E" by blast+
1941    AOT_hence ‹b = ιx φ{x}› using h "≡E" by blast
1942    AOT_thus ‹ψ{ιx φ{x}}› using j "rule=E" by blast
1943  qed
1944qed
1945end
1946
1947AOT_theorem "actual-desc:1": ‹ιx φ{x}↓ ≡ ∃!x 𝒜φ{x}›
1948proof (rule "≡I"; rule "→I")
1949  AOT_assume ‹ιx φ{x}↓›
1950  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1951  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1952  moreover AOT_have ‹a = ιx φ{x} ≡ ∀z(𝒜φ{z} ≡ z = a)›
1953    using descriptions[axiom_inst] by blast
1954  ultimately AOT_have ‹∀z(𝒜φ{z} ≡ z = a)›
1955    using "≡E" by blast
1956  AOT_hence ‹∃x∀z(𝒜φ{z} ≡ z = x)› by (rule "∃I")
1957  AOT_thus ‹∃!x 𝒜φ{x}›
1958    using "uniqueness:2"[THEN "≡E"(2)] by fast
1959next
1960  AOT_assume ‹∃!x 𝒜φ{x}›
1961  AOT_hence ‹∃x∀z(𝒜φ{z} ≡ z = x)›
1962    using "uniqueness:2"[THEN "≡E"(1)] by fast
1963  then AOT_obtain a where ‹∀z(𝒜φ{z} ≡ z = a)› using "instantiation"[rotated] by blast
1964  moreover AOT_have ‹a = ιx φ{x} ≡ ∀z(𝒜φ{z} ≡ z = a)›
1965    using descriptions[axiom_inst] by blast
1966  ultimately AOT_have ‹a = ιx φ{x}› using "≡E" by blast
1967  AOT_thus ‹ιx φ{x}↓› by (metis "t=t-proper:2" "vdash-properties:6")
1968qed
1969
1970AOT_theorem "actual-desc:2": ‹x = ιx φ{x} → 𝒜φ{x}›
1971  using "&E"(1) "contraposition:1[2]" "≡E"(1) "nec-hintikka-scheme" "reductio-aa:2" "vdash-properties:9" by blast
1972
1973AOT_theorem "actual-desc:3": ‹z = ιx φ{x} → 𝒜φ{z}›
1974  using "actual-desc:2". (* TODO: same as above *)
1975
1976AOT_theorem "actual-desc:4": ‹ιx φ{x}↓ → 𝒜φ{ιx φ{x}}›
1977proof(rule "→I")
1978  AOT_assume ‹ιx φ{x}↓›
1979  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1980  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1981  AOT_thus ‹𝒜φ{ιx φ{x}}›
1982    using "actual-desc:2" "rule=E" "→E" by fast
1983qed
1984
1985(* TODO: take another look at proof in PLM *)
1986AOT_theorem "actual-desc:5": ‹ιx φ{x} = ιx ψ{x} → 𝒜∀x(φ{x} ≡ ψ{x})›
1987proof(rule "→I")
1988  AOT_assume 0: ‹ιx φ{x} = ιx ψ{x}›
1989  AOT_hence φ_down: ‹ιx φ{x}↓› and ψ_down: ‹ιx ψ{x}↓›
1990    using "t=t-proper:1" "t=t-proper:2" "vdash-properties:6" by blast+
1991  AOT_hence ‹∃y (y = ιx φ{x})› and ‹∃y (y = ιx ψ{x})› by (metis "rule=I:1" "existential:1")+
1992  then AOT_obtain a and b where a_eq: ‹a = ιx φ{x}› and b_eq: ‹b = ιx ψ{x}›
1993    using "instantiation"[rotated] by metis
1994
1995  AOT_have ‹∀α∀β (α = β → β = α)› by (rule "∀I"; rule "∀I"; rule "id-eq:2")
1996  AOT_hence ‹∀β (ιx φ{x} = β → β = ιx φ{x})›
1997    using "∀E" φ_down by blast
1998  AOT_hence ‹ιx φ{x} = ιx ψ{x} → ιx ψ{x} = ιx φ{x}›
1999    using "∀E" ψ_down by blast
2000  AOT_hence 1: ‹ιx ψ{x} = ιx φ{x}› using 0
2001    "→E" by blast
2002
2003  AOT_have ‹𝒜φ{x} ≡ 𝒜ψ{x}› for x
2004  proof(rule "≡I"; rule "→I")
2005    AOT_assume ‹𝒜φ{x}›
2006    moreover AOT_have ‹𝒜φ{x} → x = a› for x
2007      using "nec-hintikka-scheme"[THEN "≡E"(1), OF a_eq, THEN "&E"(2)] "∀E" by blast
2008    ultimately AOT_have ‹x = a› using "→E" by blast
2009    AOT_hence ‹x = ιx φ{x}› using a_eq "rule=E" by blast
2010    AOT_hence ‹x = ιx ψ{x}› using 0 "rule=E" by blast
2011    AOT_thus ‹𝒜ψ{x}› by (metis "actual-desc:3" "vdash-properties:6")
2012  next
2013    AOT_assume ‹𝒜ψ{x}›
2014    moreover AOT_have ‹𝒜ψ{x} → x = b› for x
2015      using "nec-hintikka-scheme"[THEN "≡E"(1), OF b_eq, THEN "&E"(2)] "∀E" by blast
2016    ultimately AOT_have ‹x = b› using "→E" by blast
2017    AOT_hence ‹x = ιx ψ{x}› using b_eq "rule=E" by blast
2018    AOT_hence ‹x = ιx φ{x}› using 1 "rule=E" by blast
2019    AOT_thus ‹𝒜φ{x}› by (metis "actual-desc:3" "vdash-properties:6")
2020  qed
2021  AOT_hence ‹𝒜(φ{x} ≡ ψ{x})› for x by (metis "Act-Basic:5" "≡E"(2))
2022  AOT_hence ‹∀x 𝒜(φ{x} ≡ ψ{x})› by (rule "∀I")
2023  AOT_thus ‹𝒜∀x (φ{x} ≡ ψ{x})›
2024    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(2)] by fast
2025qed    
2026
2027AOT_theorem "!box-desc:1": ‹∃!x □φ{x} → ∀y (y = ιx φ{x} → φ{y})›
2028proof(rule "→I")
2029  AOT_assume ‹∃!x □φ{x}›
2030  AOT_hence ζ: ‹∃x (□φ{x} & ∀z (□φ{z} → z = x))›
2031    using "uniqueness:1"[THEN "≡dfE"] by blast
2032  then AOT_obtain b where θ: ‹□φ{b} & ∀z (□φ{z} → z = b)› using "instantiation"[rotated] by blast
2033  AOT_show ‹∀y (y = ιx φ{x} → φ{y})›
2034  proof(rule GEN; rule "→I")
2035    fix y
2036    AOT_assume ‹y = ιx φ{x}›
2037    AOT_hence ‹𝒜φ{y} & ∀z (𝒜φ{z} → z = y)› using "nec-hintikka-scheme"[THEN "≡E"(1)] by blast
2038    AOT_hence ‹𝒜φ{b} → b = y› using "&E" "∀E" by blast
2039    moreover AOT_have ‹𝒜φ{b}› using θ[THEN "&E"(1)]  by (metis "nec-imp-act" "→E")
2040    ultimately AOT_have ‹b = y› using "→E" by blast
2041    moreover AOT_have ‹φ{b}› using θ[THEN "&E"(1)]  by (metis "qml:2"[axiom_inst] "→E") 
2042    ultimately AOT_show ‹φ{y}› using "rule=E" by blast
2043  qed
2044qed
2045
2046AOT_theorem "!box-desc:2": ‹∀x (φ{x} → □φ{x}) → (∃!x φ{x} → ∀y (y = ιx φ{x} → φ{y}))›
2047proof(rule "→I"; rule "→I")
2048  AOT_assume ‹∀x (φ{x} → □φ{x})›
2049  moreover AOT_assume ‹∃!x φ{x}›
2050  ultimately AOT_have ‹∃!x □φ{x}›
2051    using "nec-exist-!"[THEN "→E", THEN "→E"] by blast
2052  AOT_thus ‹∀y (y = ιx φ{x} → φ{y})›
2053    using "!box-desc:1" "→E" by blast
2054qed
2055
2056AOT_theorem "dr-alphabetic-thm": ‹ιν φ{ν}↓ → ιν φ{ν} = ιμ φ{μ}› (* TODO: vacuous *)
2057  by (simp add: "rule=I:1" "→I")
2058
2059AOT_theorem "RM:1[prem]": assumes ‹Γ ⊢□ φ → ψ› shows ‹□Γ ⊢□ □φ → □ψ›
2060proof -
2061  AOT_have ‹□Γ ⊢□ □(φ → ψ)› using "RN[prem]" assms by blast
2062  AOT_thus ‹□Γ ⊢□ □φ → □ψ› by (metis "qml:1"[axiom_inst] "→E")
2063qed
2064
2065AOT_theorem "RM:1": assumes ‹⊢□ φ → ψ› shows ‹⊢□ □φ → □ψ›
2066  using "RM:1[prem]" assms by blast
2067
2068lemmas RM = "RM:1"
2069
2070AOT_theorem "RM:2[prem]": assumes ‹Γ ⊢□ φ → ψ› shows ‹□Γ ⊢□ ◇φ → ◇ψ›
2071proof -
2072  AOT_have ‹Γ ⊢□ ¬ψ → ¬φ› using assms 
2073    by (simp add: "contraposition:1[1]")
2074  AOT_hence ‹□Γ ⊢□ □¬ψ → □¬φ› using "RM:1[prem]" by blast
2075  AOT_thus ‹□Γ ⊢□ ◇φ → ◇ψ›
2076    by (meson "≡dfE" "≡dfI" "conventions:5" "deduction-theorem" "modus-tollens:1")
2077qed
2078
2079AOT_theorem "RM:2": assumes ‹⊢□ φ → ψ› shows ‹⊢□ ◇φ → ◇ψ›
2080  using "RM:2[prem]" assms by blast
2081
2082lemmas "RM◇" = "RM:2"
2083
2084AOT_theorem "RM:3[prem]": assumes ‹Γ ⊢□ φ ≡ ψ› shows ‹□Γ ⊢□ □φ ≡ □ψ›
2085proof -
2086  AOT_have ‹Γ ⊢□ φ → ψ› and ‹Γ ⊢□ ψ → φ› using assms "≡E" "→I" by metis+
2087  AOT_hence ‹□Γ ⊢□ □φ → □ψ› and ‹□Γ ⊢□ □ψ → □φ› using "RM:1[prem]" by metis+
2088  AOT_thus ‹□Γ ⊢□ □φ ≡ □ψ›
2089    by (simp add: "≡I")
2090qed
2091
2092AOT_theorem "RM:3": assumes ‹⊢□ φ ≡ ψ› shows ‹⊢□ □φ ≡ □ψ›
2093  using "RM:3[prem]" assms by blast
2094
2095lemmas RE = "RM:3"
2096
2097AOT_theorem "RM:4[prem]": assumes ‹Γ ⊢□ φ ≡ ψ› shows ‹□Γ ⊢□ ◇φ ≡ ◇ψ›
2098proof -
2099  AOT_have ‹Γ ⊢□ φ → ψ› and ‹Γ ⊢□ ψ → φ› using assms "≡E" "→I" by metis+
2100  AOT_hence ‹□Γ ⊢□ ◇φ → ◇ψ› and ‹□Γ ⊢□ ◇ψ → ◇φ› using "RM:2[prem]" by metis+
2101  AOT_thus ‹□Γ ⊢□ ◇φ ≡ ◇ψ› by (simp add: "≡I")
2102qed
2103
2104AOT_theorem "RM:4": assumes ‹⊢□ φ ≡ ψ› shows ‹⊢□ ◇φ ≡ ◇ψ›
2105  using "RM:4[prem]" assms by blast
2106
2107lemmas "RE◇" = "RM:4"
2108
2109AOT_theorem "KBasic:1": ‹□φ → □(ψ → φ)›
2110  by (simp add: RM "pl:1"[axiom_inst])
2111
2112AOT_theorem "KBasic:2": ‹□¬φ → □(φ → ψ)›
2113  by (simp add: RM "useful-tautologies:3")
2114
2115AOT_theorem "KBasic:3": ‹□(φ & ψ) ≡ (□φ & □ψ)›
2116proof (rule "≡I"; rule "→I")
2117  AOT_assume ‹□(φ & ψ)›
2118  AOT_thus ‹□φ & □ψ›
2119    by (meson RM "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "vdash-properties:6")
2120next
2121  AOT_have ‹□φ → □(ψ → (φ & ψ))› by (simp add: "RM:1" Adjunction)
2122  AOT_hence ‹□φ → (□ψ → □(φ & ψ))›  by (metis "Hypothetical Syllogism" "qml:1"[axiom_inst])
2123  moreover AOT_assume ‹□φ & □ψ›
2124  ultimately AOT_show ‹□(φ & ψ)›
2125    using "→E" "&E" by blast
2126qed
2127
2128AOT_theorem "KBasic:4": ‹□(φ ≡ ψ) ≡ (□(φ → ψ) & □(ψ → φ))›
2129proof -
2130  AOT_have θ: ‹□((φ → ψ) & (ψ → φ)) ≡ (□(φ → ψ) & □(ψ → φ))›
2131    by (fact "KBasic:3")
2132  AOT_modally_strict {
2133    AOT_have ‹(φ ≡ ψ) ≡ ((φ → ψ) & (ψ → φ))›
2134      by (fact "conventions:3"[THEN "≡Df"])
2135  }
2136  AOT_hence ξ: ‹□(φ ≡ ψ) ≡ □((φ → ψ) & (ψ → φ))›
2137    by (rule RE)
2138  with ξ and θ AOT_show ‹□(φ ≡ ψ) ≡ (□(φ → ψ) & □(ψ → φ))›
2139    using "≡E"(5) by blast
2140qed
2141
2142AOT_theorem "KBasic:5": ‹(□(φ → ψ) & □(ψ → φ)) → (□φ ≡ □ψ)›
2143proof -
2144  AOT_have ‹□(φ → ψ) → (□φ → □ψ)›
2145    by (fact "qml:1"[axiom_inst])
2146  moreover AOT_have ‹□(ψ → φ) → (□ψ → □φ)›
2147    by (fact "qml:1"[axiom_inst])
2148  ultimately AOT_have ‹(□(φ → ψ) & □(ψ → φ)) → ((□φ → □ψ) & (□ψ → □φ))›
2149    by (metis "&I" MP "Double Composition")
2150  moreover AOT_have ‹((□φ → □ψ) & (□ψ → □φ)) → (□φ ≡ □ψ)›
2151    using "conventions:3"[THEN "≡dfI"] "→I" by blast
2152  ultimately AOT_show ‹(□(φ → ψ) & □(ψ → φ)) → (□φ ≡ □ψ)›
2153    by (metis "Hypothetical Syllogism")
2154qed
2155
2156AOT_theorem "KBasic:6": ‹□(φ≡ ψ) → (□φ ≡ □ψ)›
2157  using "KBasic:4" "KBasic:5" "deduction-theorem" "≡E"(1) "vdash-properties:10" by blast
2158AOT_theorem "KBasic:7": ‹((□φ & □ψ) ∨ (□¬φ & □¬ψ)) → □(φ ≡ ψ)›
2159proof (rule "→I"; drule "∨E"(1); (rule "→I")?)
2160  AOT_assume ‹□φ & □ψ›
2161  AOT_hence ‹□φ› and ‹□ψ› using "&E" by blast+
2162  AOT_hence ‹□(φ → ψ)› and ‹□(ψ → φ)› using "KBasic:1" "→E" by blast+
2163  AOT_hence ‹□(φ → ψ) & □(ψ → φ)› using "&I" by blast
2164  AOT_thus ‹□(φ ≡ ψ)›  by (metis "KBasic:4" "≡E"(2))
2165next
2166  AOT_assume ‹□¬φ & □¬ψ›
2167  AOT_hence 0: ‹□(¬φ & ¬ψ)› using "KBasic:3"[THEN "≡E"(2)] by blast
2168  AOT_modally_strict {
2169    AOT_have ‹(¬φ & ¬ψ) → (φ ≡ ψ)›
2170      by (metis "&E"(1) "&E"(2) "deduction-theorem" "≡I" "reductio-aa:1")
2171  }
2172  AOT_hence ‹□(¬φ & ¬ψ) → □(φ ≡ ψ)›
2173    by (rule RM)
2174  AOT_thus ‹□(φ ≡ ψ)› using 0 "→E" by blast
2175qed(auto)
2176
2177AOT_theorem "KBasic:8": ‹□(φ & ψ) → □(φ ≡ ψ)›
2178  by (meson "RM:1" "&E"(1) "&E"(2) "deduction-theorem" "≡I")
2179AOT_theorem "KBasic:9": ‹□(¬φ & ¬ψ) → □(φ ≡ ψ)›
2180  by (metis "RM:1" "&E"(1) "&E"(2) "deduction-theorem" "≡I" "raa-cor:4")
2181AOT_theorem "KBasic:10": ‹□φ ≡ □¬¬φ›
2182  by (simp add: "RM:3" "oth-class-taut:3:b")
2183AOT_theorem "KBasic:11": ‹¬□φ ≡ ◇¬φ›
2184proof (rule "≡I"; rule "→I")
2185  AOT_show ‹◇¬φ› if ‹¬□φ›
2186    using that "≡dfI" "conventions:5" "KBasic:10" "≡E"(3) by blast
2187next
2188  AOT_show ‹¬□φ› if ‹◇¬φ›
2189    using "≡dfE" "conventions:5" "KBasic:10" "≡E"(4) that by blast
2190qed
2191AOT_theorem "KBasic:12": ‹□φ ≡ ¬◇¬φ›
2192proof (rule "≡I"; rule "→I")
2193  AOT_show ‹¬◇¬φ› if ‹□φ›
2194    using "¬¬I" "KBasic:11" "≡E"(3) that by blast
2195next
2196  AOT_show ‹□φ› if ‹¬◇¬φ›
2197  using "KBasic:11" "≡E"(1) "reductio-aa:1" that by blast
2198qed
2199AOT_theorem "KBasic:13": ‹□(φ → ψ) → (◇φ → ◇ψ)›
2200proof -
2201  AOT_have ‹φ → ψ ⊢□ φ → ψ› by blast
2202  AOT_hence ‹□(φ → ψ) ⊢□ ◇φ → ◇ψ›
2203    using "RM:2[prem]" by blast
2204  AOT_thus ‹□(φ → ψ) → (◇φ → ◇ψ)› using "→I" by blast
2205qed
2206lemmas "K◇" = "KBasic:13"
2207AOT_theorem "KBasic:14": ‹◇□φ ≡ ¬□◇¬φ›
2208  by (meson "RE◇" "KBasic:11" "KBasic:12" "≡E"(6) "oth-class-taut:3:a")
2209AOT_theorem "KBasic:15": ‹(□φ ∨ □ψ) → □(φ ∨ ψ)›
2210proof -
2211  AOT_modally_strict {
2212    AOT_have ‹φ → (φ ∨ ψ)› and ‹ψ → (φ ∨ ψ)›
2213      by (auto simp: "Disjunction Addition"(1) "Disjunction Addition"(2))
2214  }
2215  AOT_hence ‹□φ → □(φ ∨ ψ)› and ‹□ψ → □(φ ∨ ψ)›
2216    using RM by blast+
2217  AOT_thus ‹(□φ ∨ □ψ) → □(φ ∨ ψ)›
2218    by (metis "∨E"(1) "deduction-theorem")
2219qed
2220
2221AOT_theorem "KBasic:16": ‹(□φ & ◇ψ) → ◇(φ & ψ)›
2222  by (meson "KBasic:13" "RM:1" Adjunction "Hypothetical Syllogism" Importation "vdash-properties:6")
2223
2224AOT_theorem "rule-sub-lem:1:a":
2225  assumes ‹⊢□ □(ψ ≡ χ)›
2226  shows ‹⊢□ ¬ψ ≡ ¬χ›
2227  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2228        "≡E"(1) "oth-class-taut:4:b" by blast
2229
2230AOT_theorem "rule-sub-lem:1:b":
2231  assumes ‹⊢□ □(ψ ≡ χ)›
2232  shows ‹⊢□ (ψ → Θ) ≡ (χ → Θ)›
2233  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2234  using "oth-class-taut:4:c" "vdash-properties:6" by blast
2235
2236AOT_theorem "rule-sub-lem:1:c":
2237  assumes ‹⊢□ □(ψ ≡ χ)›
2238  shows ‹⊢□ (Θ → ψ) ≡ (Θ → χ)›
2239  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2240  using "oth-class-taut:4:d" "vdash-properties:6" by blast
2241
2242AOT_theorem "rule-sub-lem:1:d":
2243  assumes ‹for arbitrary α: ⊢□ □(ψ{α} ≡ χ{α})›
2244  shows ‹⊢□ ∀α ψ{α} ≡ ∀α χ{α}›
2245proof -
2246  AOT_modally_strict {
2247    AOT_have ‹∀α (ψ{α} ≡ χ{α})›
2248      using "qml:2"[axiom_inst, THEN "→E", OF assms] "∀I" by fast
2249    AOT_hence 0: ‹ψ{α} ≡ χ{α}› for α using "∀E" by blast
2250    AOT_show ‹∀α ψ{α} ≡ ∀α χ{α}›
2251    proof (rule "≡I"; rule "→I")
2252      AOT_assume ‹∀α ψ{α}›
2253      AOT_hence ‹ψ{α}› for α using "∀E" by blast
2254      AOT_hence ‹χ{α}› for α using 0 "≡E" by blast
2255      AOT_thus ‹∀α χ{α}› by (rule "∀I")
2256    next
2257      AOT_assume ‹∀α χ{α}›
2258      AOT_hence ‹χ{α}› for α using "∀E" by blast
2259      AOT_hence ‹ψ{α}› for α using 0 "≡E" by blast
2260      AOT_thus ‹∀α ψ{α}› by (rule "∀I")
2261    qed
2262  }
2263qed
2264
2265AOT_theorem "rule-sub-lem:1:e":
2266  assumes ‹⊢□ □(ψ ≡ χ)›
2267  shows ‹⊢□ [λ ψ] ≡ [λ χ]›
2268  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2269  using "≡E"(1) "propositions-lemma:6" by blast
2270
2271AOT_theorem "rule-sub-lem:1:f":
2272  assumes ‹⊢□ □(ψ ≡ χ)›
2273  shows ‹⊢□ 𝒜ψ ≡ 𝒜χ›
2274  using "qml:2"[axiom_inst, THEN "→E", OF assms, THEN "RA[2]"]
2275  by (metis "Act-Basic:5" "≡E"(1))
2276
2277AOT_theorem "rule-sub-lem:1:g":
2278  assumes ‹⊢□ □(ψ ≡ χ)›
2279  shows ‹⊢□ □ψ ≡ □χ›
2280  using "KBasic:6" assms "vdash-properties:6" by blast
2281
2282text‹Note that instead of deriving @{text "rule-sub-lem:2"}, @{text "rule-sub-lem:3"}, @{text "rule-sub-lem:4"},
2283     and @{text "rule-sub-nec"}, we construct substitution methods instead.›
2284
2285class AOT_subst =
2286  fixes AOT_subst :: "('a ⇒ 𝗈) ⇒ bool"
2287    and AOT_subst_cond :: "'a ⇒ 'a ⇒ bool"
2288  assumes AOT_subst: "AOT_subst φ ⟹ AOT_subst_cond ψ χ ⟹ [v ⊨ «φ ψ» ≡ «φ χ»]"
2289
2290named_theorems AOT_substI
2291
2292instantiation 𝗈 :: AOT_subst
2293begin
2294
2295inductive AOT_subst_𝗈 where
2296  AOT_subst_𝗈_id[AOT_substI]: "AOT_subst_𝗈 (λφ. φ)"
2297| AOT_subst_𝗈_const[AOT_substI]: "AOT_subst_𝗈 (λφ. ψ)"
2298| AOT_subst_𝗈_not[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «¬Θ{φ}»)"
2299| AOT_subst_𝗈_imp[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 Ξ ⟹ AOT_subst_𝗈 (λ φ. «Θ{φ} → Ξ{φ}»)"
2300| AOT_subst_𝗈_lambda0[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. (AOT_lambda0 (Θ φ)))"
2301| AOT_subst_𝗈_act[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «𝒜Θ{φ}»)"
2302| AOT_subst_𝗈_box[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «□Θ{φ}»)"
2303| AOT_subst_𝗈_by_def[AOT_substI]: "(⋀ ψ . AOT_model_equiv_def (Θ ψ) (Ξ ψ)) ⟹ AOT_subst_𝗈 Ξ ⟹ AOT_subst_𝗈 Θ"
2304
2305definition AOT_subst_cond_𝗈 where "AOT_subst_cond_𝗈 ≡ λ ψ χ . ∀ v . [v ⊨ ψ ≡ χ]"
2306
2307instance
2308proof
2309  fix ψ χ :: 𝗈 and φ :: ‹𝗈 ⇒ 𝗈›
2310  assume cond: ‹AOT_subst_cond ψ χ›
2311  assume ‹AOT_subst φ›
2312  moreover AOT_have ‹⊢□ ψ ≡ χ› using cond unfolding AOT_subst_cond_𝗈_def by blast
2313  ultimately AOT_show ‹⊢□ φ{ψ} ≡ φ{χ}›
2314  proof (induct arbitrary: ψ χ)
2315    case AOT_subst_𝗈_id
2316    thus ?case using "≡E"(2) "oth-class-taut:4:b" "rule-sub-lem:1:a" by blast
2317  next
2318    case (AOT_subst_𝗈_const ψ)
2319    thus ?case by (simp add: "oth-class-taut:3:a")
2320  next
2321    case (AOT_subst_𝗈_not Θ)
2322    thus ?case by (simp add: RN "rule-sub-lem:1:a")
2323  next
2324    case (AOT_subst_𝗈_imp Θ Ξ)
2325    thus ?case by (meson RN "≡E"(5) "rule-sub-lem:1:b" "rule-sub-lem:1:c")
2326  next
2327    case (AOT_subst_𝗈_lambda0 Θ)
2328    thus ?case by (simp add: RN "rule-sub-lem:1:e")
2329  next
2330    case (AOT_subst_𝗈_act Θ)
2331    thus ?case by (simp add: RN "rule-sub-lem:1:f")
2332  next
2333    case (AOT_subst_𝗈_box Θ)
2334    thus ?case by (simp add: RN "rule-sub-lem:1:g")
2335  next
2336    case (AOT_subst_𝗈_by_def Θ Ξ)
2337    AOT_modally_strict {
2338      AOT_have ‹Ξ{ψ} ≡ Ξ{χ}› using AOT_subst_𝗈_by_def by simp
2339      AOT_thus ‹Θ{ψ} ≡ Θ{χ}›
2340        using "≡Df"[OF AOT_subst_𝗈_by_def(1), of _ ψ] "≡Df"[OF AOT_subst_𝗈_by_def(1), of _ χ]
2341        by (metis "≡E"(6) "oth-class-taut:3:a")
2342    }
2343  qed
2344qed
2345end
2346
2347instantiation "fun" :: (AOT_Term_id_2, AOT_subst) AOT_subst
2348begin
2349
2350definition AOT_subst_cond_fun :: "('a ⇒ 'b) ⇒ ('a ⇒ 'b) ⇒ bool" where
2351  "AOT_subst_cond_fun ≡ λ φ ψ . ∀ α . AOT_subst_cond (φ (AOT_term_of_var α)) (ψ (AOT_term_of_var α))"
2352
2353inductive AOT_subst_fun :: "(('a ⇒ 'b) ⇒ 𝗈) ⇒ bool" where
2354  AOT_subst_fun_const[AOT_substI]: "AOT_subst_fun (λφ. ψ)"
2355| AOT_subst_fun_id[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst_fun (λφ. Ψ (φ (AOT_term_of_var x)))"
2356| AOT_subst_fun_all[AOT_substI]: "AOT_subst Ψ ⟹ (⋀ α . AOT_subst_fun (Θ (AOT_term_of_var α))) ⟹ AOT_subst_fun (λφ :: 'a ⇒ 'b. Ψ «∀α «Θ (α::'a) φ»»)"
2357| AOT_subst_fun_not[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst_fun (λφ. «¬«Ψ φ»»)"
2358| AOT_subst_fun_imp[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst Θ ⟹ AOT_subst_fun (λφ. ««Ψ φ» → «Θ φ»»)"
2359| AOT_subst_fun_lambda0[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. (AOT_lambda0 (Θ φ)))"
2360| AOT_subst_fun_act[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. «𝒜«Θ φ»»)"
2361| AOT_subst_fun_box[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. «□«Θ φ»»)"
2362| AOT_subst_fun_def[AOT_substI]: "(⋀ φ . AOT_model_equiv_def (Θ φ) (Ψ φ)) ⟹ AOT_subst_fun Ψ ⟹ AOT_subst_fun Θ"
2363
2364instance proof
2365  fix ψ χ :: "'a ⇒ 'b" and φ :: ‹('a ⇒ 'b) ⇒ 𝗈›
2366  assume ‹AOT_subst φ›
2367  moreover assume cond: ‹AOT_subst_cond ψ χ›
2368  ultimately AOT_show ‹⊢□ «φ ψ» ≡ «φ χ»›
2369  proof(induct)
2370    case (AOT_subst_fun_const ψ)
2371    then show ?case by (simp add: "oth-class-taut:3:a")
2372  next
2373  case (AOT_subst_fun_id Ψ x)
2374  then show ?case by (simp add: AOT_subst AOT_subst_cond_fun_def) 
2375  next
2376  case (AOT_subst_fun_all Ψ Θ)
2377  AOT_have ‹⊢□ □(Θ{α, «ψ»} ≡ Θ{α, «χ»})› for α
2378    using AOT_subst_fun_all.hyps(3) AOT_subst_fun_all.prems RN by presburger
2379  thus ?case using AOT_subst[OF AOT_subst_fun_all(1)]
2380    by (simp add: RN "rule-sub-lem:1:d" AOT_subst_cond_fun_def AOT_subst_cond_𝗈_def)
2381  next
2382  case (AOT_subst_fun_not Ψ)
2383  then show ?case by (simp add: RN "rule-sub-lem:1:a")
2384  next
2385  case (AOT_subst_fun_imp Ψ Θ)
2386  then show ?case 
2387    unfolding AOT_subst_cond_fun_def AOT_subst_cond_𝗈_def
2388    by (meson "≡E"(5) "oth-class-taut:4:c" "oth-class-taut:4:d" "vdash-properties:6")
2389  next
2390  case (AOT_subst_fun_lambda0 Θ)
2391  then show ?case by (simp add: RN "rule-sub-lem:1:e")
2392  next
2393  case (AOT_subst_fun_act Θ)
2394  then show ?case by (simp add: RN "rule-sub-lem:1:f")
2395  next
2396  case (AOT_subst_fun_box Θ)
2397  then show ?case by (simp add: RN "rule-sub-lem:1:g")
2398  next
2399  case (AOT_subst_fun_def Θ Ψ)
2400  then show ?case
2401    by (meson "df-rules-formulas[3]" "df-rules-formulas[4]" "≡I" "≡E"(5))
2402  qed
2403qed
2404end
2405
2406method_setup AOT_defI =
2407‹Scan.lift (Scan.succeed (fn ctxt => (Method.CONTEXT_METHOD (fn thms => (Context_Tactic.CONTEXT_SUBGOAL (fn (trm,int) => 
2408Context_Tactic.CONTEXT_TACTIC (
2409let
2410fun findHeadConst (Const x) = SOME x
2411  | findHeadConst (A $ B) = findHeadConst A
2412  | findHeadConst _ = NONE
2413fun findDef (Const (const_name‹AOT_model_equiv_def›, _) $ lhs $ rhs) = findHeadConst lhs
2414  | findDef (A $ B) = (case findDef A of SOME x => SOME x | _ => findDef B)
2415  | findDef (Abs (a,b,c)) = findDef c
2416  | findDef _ = NONE
2417val const_opt = (findDef trm)
2418val defs = case const_opt of SOME const => List.filter (fn thm => let
2419    val concl = Thm.concl_of thm
2420    val thmconst = (findDef concl)
2421    in case thmconst of SOME (c,_) => fst const = c | _ => false end) (AOT_Definitions.get ctxt)
2422    | _ => []
2423in
2424resolve_tac ctxt defs 1
2425end
2426)) 1)))))›
2427‹Resolve AOT definitions›
2428
2429method AOT_subst_intro_helper = ((rule AOT_substI
2430      | AOT_defI
2431      | (simp only: AOT_subst_cond_𝗈_def AOT_subst_cond_fun_def; ((rule allI)+)?)))
2432
2433method AOT_subst for ψ::"'a::AOT_subst" and χ::"'a::AOT_subst" =
2434    (match conclusion in "[v ⊨ «φ ψ»]" for φ and v ⇒
2435      ‹match (φ) in "λa . ?p" ⇒ ‹fail› ¦ "λa . a" ⇒ ‹fail›
2436       ¦ _ ⇒ ‹rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(2)]
2437       ; (AOT_subst_intro_helper+)?››)
2438
2439method AOT_subst_rev for χ::"'a::AOT_subst" and ψ::"'a::AOT_subst" =
2440    (match conclusion in "[v ⊨ «φ ψ»]" for φ and v ⇒
2441      ‹match (φ) in "λa . ?p" ⇒ ‹fail› ¦ "λa . a" ⇒ ‹fail›
2442       ¦ _ ⇒ ‹rule AOT_subst[where φ=φ and ψ=χ and χ=ψ, THEN "≡E"(1)]
2443       ; (AOT_subst_intro_helper+)?››)
2444
2445method AOT_subst_manual for φ::"'a::AOT_subst ⇒ 𝗈" =
2446    (rule AOT_subst[where φ=φ, THEN "≡E"(2)]; (AOT_subst_intro_helper+)?)
2447
2448method AOT_subst_manual_rev for φ::"'a::AOT_subst ⇒ 𝗈" =
2449    (rule AOT_subst[where φ=φ, THEN "≡E"(1)]; (AOT_subst_intro_helper+)?)
2450
2451method AOT_subst_using uses subst =
2452    (match subst in "[?w ⊨ ψ ≡ χ]" for ψ χ ⇒ ‹
2453       match conclusion in "[v ⊨ «φ ψ»]" for φ v ⇒ ‹
2454         rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(2)]
2455         ; ((AOT_subst_intro_helper | (fact subst; fail))+)?››)
2456
2457method AOT_subst_using_rev uses subst =
2458    (match subst in "[?w ⊨ ψ ≡ χ]" for ψ χ ⇒ ‹
2459      match conclusion in "[v ⊨ «φ χ»]" for φ v ⇒ ‹
2460        rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(1)]
2461        ; ((AOT_subst_intro_helper | (fact subst; fail))+)?››)
2462
2463AOT_theorem "rule-sub-remark:1[1]": assumes ‹⊢□ A!x ≡ ¬◇E!x› and ‹¬A!x› shows ‹¬¬◇E!x›
2464  by (AOT_subst_rev "«A!x»" "«¬◇E!x»") (auto simp: assms)
2465
2466AOT_theorem "rule-sub-remark:1[2]": assumes ‹⊢□ A!x ≡ ¬◇E!x› and  ‹¬¬◇E!x› shows ‹¬A!x›
2467  by (AOT_subst "«A!x»" "«¬◇E!x»") (auto simp: assms)
2468
2469AOT_theorem "rule-sub-remark:2[1]":
2470  assumes ‹⊢□ [R]xy ≡ ([R]xy & ([Q]a ∨ ¬[Q]a))› and ‹p → [R]xy› shows ‹p → [R]xy & ([Q]a ∨ ¬[Q]a)›
2471  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2472
2473AOT_theorem "rule-sub-remark:2[2]":
2474  assumes ‹⊢□ [R]xy ≡ ([R]xy & ([Q]a ∨ ¬[Q]a))› and ‹p → [R]xy & ([Q]a ∨ ¬[Q]a)› shows ‹p → [R]xy›
2475  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2476
2477AOT_theorem "rule-sub-remark:3[1]":
2478  assumes ‹for arbitrary x: ⊢□ A!x ≡ ¬◇E!x›
2479      and ‹∃x A!x›
2480    shows ‹∃x ¬◇E!x›
2481  by (AOT_subst_rev "λκ. «A!κ»" "λκ. «¬◇E!κ»") (auto simp: assms)
2482
2483AOT_theorem "rule-sub-remark:3[2]":
2484  assumes ‹for arbitrary x: ⊢□ A!x ≡ ¬◇E!x›
2485      and ‹∃x ¬◇E!x›
2486    shows ‹∃x A!x›
2487  by (AOT_subst "λκ. «A!κ»" "λκ. «¬◇E!κ»") (auto simp: assms)
2488
2489AOT_theorem "rule-sub-remark:4[1]":
2490  assumes ‹⊢□ ¬¬[P]x ≡ [P]x› and ‹𝒜¬¬[P]x› shows ‹𝒜[P]x›
2491  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2492
2493AOT_theorem "rule-sub-remark:4[2]":
2494  assumes ‹⊢□ ¬¬[P]x ≡ [P]x› and ‹𝒜[P]x› shows ‹𝒜¬¬[P]x›
2495  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2496
2497AOT_theorem "rule-sub-remark:5[1]":
2498  assumes ‹⊢□ (φ → ψ) ≡ (¬ψ → ¬φ)› and ‹□(φ → ψ)› shows ‹□(¬ψ → ¬φ)›
2499  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2500
2501AOT_theorem "rule-sub-remark:5[2]":
2502  assumes ‹⊢□ (φ → ψ) ≡ (¬ψ → ¬φ)› and ‹□(¬ψ → ¬φ)› shows ‹□(φ → ψ)› 
2503  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2504
2505AOT_theorem "rule-sub-remark:6[1]":
2506  assumes ‹⊢□ ψ ≡ χ› and ‹□(φ → ψ)› shows ‹□(φ → χ)› 
2507  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2508
2509AOT_theorem "rule-sub-remark:6[2]":
2510  assumes ‹⊢□ ψ ≡ χ› and ‹□(φ → χ)› shows ‹□(φ → ψ)›
2511  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2512
2513AOT_theorem "rule-sub-remark:7[1]":
2514  assumes ‹⊢□ φ ≡ ¬¬φ› and ‹□(φ → φ)› shows ‹□(¬¬φ → φ)› 
2515  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2516
2517AOT_theorem "rule-sub-remark:7[2]":
2518  assumes ‹⊢□ φ ≡ ¬¬φ› and ‹□(¬¬φ → φ)› shows  ‹□(φ → φ)›
2519  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2520
2521AOT_theorem "KBasic2:1": ‹□¬φ ≡ ¬◇φ›
2522  by (meson "conventions:5" "contraposition:2" "Hypothetical Syllogism" "df-rules-formulas[3]"
2523            "df-rules-formulas[4]" "≡I" "useful-tautologies:1")
2524
2525AOT_theorem "KBasic2:2": ‹◇(φ ∨ ψ) ≡ (◇φ ∨ ◇ψ)›
2526proof -
2527  AOT_have ‹◇(φ ∨ ψ) ≡ ◇¬(¬φ & ¬ψ)›
2528    by (simp add: "RE◇" "oth-class-taut:5:b")
2529  also AOT_have ‹… ≡ ¬□(¬φ & ¬ψ)›
2530    using "KBasic:11" "≡E"(6) "oth-class-taut:3:a" by blast
2531  also AOT_have ‹… ≡ ¬(□¬φ & □¬ψ)›
2532    using "KBasic:3" "≡E"(1) "oth-class-taut:4:b" by blast
2533  also AOT_have ‹… ≡ ¬(¬◇φ & ¬◇ψ)›
2534    apply (AOT_subst_rev "«□¬φ»" "«¬◇φ»")
2535    apply (simp add: "KBasic2:1")
2536    apply (AOT_subst_rev "«□¬ψ»" "«¬◇ψ»")
2537    by (auto simp: "KBasic2:1" "oth-class-taut:3:a")
2538  also AOT_have ‹… ≡ ¬¬(◇φ ∨ ◇ψ)›
2539    using "≡E"(6) "oth-class-taut:3:b" "oth-class-taut:5:b" by blast
2540  also AOT_have ‹… ≡ ◇φ ∨ ◇ψ›
2541    by (simp add: "≡I" "useful-tautologies:1" "useful-tautologies:2")
2542  finally show ?thesis .
2543qed
2544
2545AOT_theorem "KBasic2:3": ‹◇(φ & ψ) → (◇φ & ◇ψ)›
2546  by (metis "RM◇" "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "deduction-theorem" "modus-tollens:1" "reductio-aa:1")
2547
2548AOT_theorem "KBasic2:4": ‹◇(φ → ψ) ≡ (□φ → ◇ψ)›
2549proof -
2550  AOT_have ‹◇(φ → ψ) ≡ ◇(¬φ ∨ ψ)›
2551    by (AOT_subst "«φ → ψ»" "«¬φ ∨ ψ»")
2552       (auto simp: "oth-class-taut:1:c" "oth-class-taut:3:a")
2553  also AOT_have ‹... ≡ ◇¬φ ∨ ◇ψ›
2554    by (simp add: "KBasic2:2")
2555  also AOT_have ‹... ≡ ¬□φ ∨ ◇ψ›
2556    by (AOT_subst "«¬□φ»" "«◇¬φ»")
2557       (auto simp: "KBasic:11" "oth-class-taut:3:a")
2558  also AOT_have ‹... ≡ □φ → ◇ψ›
2559    using "≡E"(6) "oth-class-taut:1:c" "oth-class-taut:3:a" by blast
2560  finally show ?thesis .
2561qed
2562
2563AOT_theorem "KBasic2:5": ‹◇◇φ ≡ ¬□□¬φ›
2564  apply (AOT_subst "«◇φ»" "«¬□¬φ»")
2565   apply (simp add: "conventions:5" "≡Df")
2566  apply (AOT_subst "«◇¬□¬φ»" "«¬□¬¬□¬φ»")
2567   apply (simp add: "conventions:5" "≡Df")
2568  apply (AOT_subst_rev "«□¬φ»"  "«¬¬□¬φ»")
2569   apply (simp add: "oth-class-taut:3:b")
2570  by (simp add: "oth-class-taut:3:a")
2571
2572
2573AOT_theorem "KBasic2:6": ‹□(φ ∨ ψ) → (□φ ∨ ◇ψ)›
2574proof(rule "→I"; rule "raa-cor:1")
2575  AOT_assume ‹□(φ ∨ ψ)›
2576  AOT_hence ‹□(¬φ → ψ)›
2577    apply - apply (AOT_subst_rev "«φ ∨ ψ»" "«¬φ → ψ»")
2578    by (simp add: "conventions:2" "≡Df")
2579  AOT_hence 1: ‹◇¬φ → ◇ψ› using "KBasic:13" "vdash-properties:10" by blast
2580  AOT_assume ‹¬(□φ ∨ ◇ψ)›
2581  AOT_hence ‹¬□φ› and ‹¬◇ψ› using "&E" "≡E"(1) "oth-class-taut:5:d" by blast+
2582  AOT_thus ‹◇ψ & ¬◇ψ› using "&I"(1) 1[THEN "→E"] "KBasic:11" "≡E"(4) "raa-cor:3" by blast
2583qed
2584
2585AOT_theorem "KBasic2:7": ‹(□(φ ∨ ψ) & ◇¬φ) → ◇ψ›
2586proof(rule "→I"; frule "&E"(1); drule "&E"(2))
2587  AOT_assume ‹□(φ ∨ ψ)›
2588  AOT_hence 1: ‹□φ ∨ ◇ψ›
2589    using "KBasic2:6" "∨I"(2) "∨E"(1) by blast
2590  AOT_assume ‹◇¬φ›
2591  AOT_hence ‹¬□φ› using "KBasic:11" "≡E"(2) by blast
2592  AOT_thus ‹◇ψ› using 1 "∨E"(2) by blast
2593qed
2594
2595AOT_theorem "T-S5-fund:1": ‹φ → ◇φ›
2596  by (meson "≡dfI" "conventions:5" "contraposition:2" "Hypothetical Syllogism" "deduction-theorem" "qml:2"[axiom_inst])
2597lemmas "T◇" = "T-S5-fund:1"
2598
2599AOT_theorem "T-S5-fund:2": ‹◇□φ → □φ›
2600proof(rule "→I")
2601  AOT_assume ‹◇□φ›
2602  AOT_hence ‹¬□◇¬φ›
2603    using "KBasic:14" "≡E"(4) "raa-cor:3" by blast
2604  moreover AOT_have ‹◇¬φ → □◇¬φ›
2605    by (fact "qml:3"[axiom_inst])
2606  ultimately AOT_have ‹¬◇¬φ›
2607    using "modus-tollens:1" by blast
2608  AOT_thus ‹□φ› using "KBasic:12" "≡E"(2) by blast
2609qed
2610lemmas "5◇" = "T-S5-fund:2"
2611
2612(* Also interestingly none of these have proofs in PLM. *)
2613AOT_theorem "Act-Sub:1": ‹𝒜φ ≡ ¬𝒜¬φ›
2614  by (AOT_subst "«𝒜¬φ»" "«¬𝒜φ»")
2615     (auto simp: "logic-actual-nec:1"[axiom_inst] "oth-class-taut:3:b")
2616
2617AOT_theorem "Act-Sub:2": ‹◇φ ≡ 𝒜◇φ›
2618  apply (AOT_subst "«◇φ»" "«¬□¬φ»")
2619   apply (simp add: "conventions:5" "≡Df")
2620  by (metis "deduction-theorem" "≡I" "≡E"(1) "≡E"(2) "≡E"(3)
2621            "logic-actual-nec:1"[axiom_inst] "qml-act:2"[axiom_inst])
2622
2623AOT_theorem "Act-Sub:3": ‹𝒜φ → ◇φ›
2624  apply (AOT_subst "«◇φ»" "«¬□¬φ»")
2625   apply (simp add: "conventions:5" "≡Df")
2626  by (metis "Act-Sub:1" "deduction-theorem" "≡E"(4) "nec-imp-act" "reductio-aa:2" "vdash-properties:6")
2627
2628
2629AOT_theorem "Act-Sub:4": ‹𝒜φ ≡ ◇𝒜φ›
2630proof (rule "≡I"; rule "→I")
2631  AOT_assume ‹𝒜φ›
2632  AOT_thus ‹◇𝒜φ› using "T◇" "vdash-properties:10" by blast
2633next
2634  AOT_assume ‹◇𝒜φ›
2635  AOT_hence ‹¬□¬𝒜φ›
2636    using "≡dfE" "conventions:5" by blast
2637  AOT_hence ‹¬□𝒜¬φ›
2638    apply - apply (AOT_subst "«𝒜¬φ»" "«¬𝒜φ»")
2639    by (simp add: "logic-actual-nec:1"[axiom_inst])
2640  AOT_thus ‹𝒜φ›
2641      using "Act-Basic:1" "Act-Basic:6" "∨E"(3) "≡E"(4) "reductio-aa:1" by blast
2642qed
2643
2644AOT_theorem "Act-Sub:5": ‹◇𝒜φ → 𝒜◇φ›
2645  by (metis "Act-Sub:2" "Act-Sub:3" "Act-Sub:4" "deduction-theorem" "≡E"(1) "≡E"(2) "vdash-properties:6")
2646
2647AOT_theorem "S5Basic:1": ‹◇φ ≡ □◇φ›
2648  by (simp add: "≡I" "qml:2" "qml:3" "vdash-properties:1[2]")
2649
2650AOT_theorem "S5Basic:2": ‹□φ ≡ ◇□φ›
2651  by (simp add: "T◇" "5◇" "≡I")
2652
2653AOT_theorem "S5Basic:3": ‹φ → □◇φ›
2654  using "T◇" "Hypothetical Syllogism" "qml:3" "vdash-properties:1[2]" by blast
2655lemmas "B" = "S5Basic:3"
2656
2657AOT_theorem "S5Basic:4": ‹◇□φ → φ›
2658  using "5◇" "Hypothetical Syllogism" "qml:2" "vdash-properties:1[2]" by blast
2659lemmas "B◇" = "S5Basic:4"
2660
2661AOT_theorem "S5Basic:5": ‹□φ → □□φ›
2662  using "RM:1" "B" "5◇" "Hypothetical Syllogism" by blast
2663lemmas "4" = "S5Basic:5"
2664
2665AOT_theorem "S5Basic:6": ‹□φ ≡ □□φ›
2666  by (simp add: "4" "≡I" "qml:2"[axiom_inst])
2667
2668AOT_theorem "S5Basic:7": ‹◇◇φ → ◇φ›
2669  apply (AOT_subst "«◇◇φ»" "«¬□¬◇φ»")
2670   apply (simp add: "conventions:5" "≡Df")
2671  apply (AOT_subst "«◇φ»" "«¬□¬φ»")
2672   apply (simp add: "conventions:5" "≡Df")
2673  apply (AOT_subst_rev "«□¬φ»" "«¬¬□¬φ»")
2674   apply (simp add: "oth-class-taut:3:b")
2675  apply (AOT_subst_rev "«□¬φ»" "«□□¬φ»")
2676   apply (simp add: "S5Basic:6")
2677  by (simp add: "if-p-then-p")
2678
2679lemmas "4◇" = "S5Basic:7"
2680
2681AOT_theorem "S5Basic:8": ‹◇◇φ ≡ ◇φ›
2682  by (simp add: "4◇" "T◇" "≡I")
2683
2684AOT_theorem "S5Basic:9": ‹□(φ ∨ □ψ) ≡ (□φ ∨ □ψ)›
2685  apply (rule "≡I"; rule "→I")
2686  using "KBasic2:6" "5◇" "∨I"(3) "if-p-then-p" "vdash-properties:10" apply blast
2687  by (meson "KBasic:15" "4" "∨I"(3) "∨E"(1) "Disjunction Addition"(1) "con-dis-taut:7"
2688            "intro-elim:1" "Commutativity of ∨")
2689
2690AOT_theorem "S5Basic:10": ‹□(φ ∨ ◇ψ) ≡ (□φ ∨ ◇ψ)›
2691(* Note: nicely this proof is entirely sledgehammer generated *)
2692proof(rule "≡I"; rule "→I")
2693  AOT_assume ‹□(φ ∨ ◇ψ)›
2694  AOT_hence ‹□φ ∨ ◇◇ψ›
2695    by (meson "KBasic2:6" "∨I"(2) "∨E"(1))
2696  AOT_thus ‹□φ ∨ ◇ψ›
2697    by (meson "B◇" "4" "4◇" "T◇" "∨I"(3))
2698next
2699  AOT_assume ‹□φ ∨ ◇ψ›
2700  AOT_hence ‹□φ ∨ □◇ψ›
2701    by (meson "S5Basic:1" "B◇" "S5Basic:6" "T◇" "5◇" "∨I"(3) "intro-elim:1")
2702  AOT_thus ‹□(φ ∨ ◇ψ)›
2703    by (meson "KBasic:15" "∨I"(3) "∨E"(1) "Disjunction Addition"(1) "Disjunction Addition"(2))
2704qed
2705
2706AOT_theorem "S5Basic:11": ‹◇(φ & ◇ψ) ≡ (◇φ & ◇ψ)›
2707proof -
2708  AOT_have ‹◇(φ & ◇ψ) ≡ ◇¬(¬φ ∨ ¬◇ψ)›
2709    by (AOT_subst "«φ & ◇ψ»" "«¬(¬φ ∨ ¬◇ψ)»")
2710       (auto simp: "oth-class-taut:5:a" "oth-class-taut:3:a")
2711  also AOT_have ‹… ≡ ◇¬(¬φ ∨ □¬ψ)›
2712    by (AOT_subst "«□¬ψ»" "«¬◇ψ»")
2713       (auto simp: "KBasic2:1" "oth-class-taut:3:a")
2714  also AOT_have ‹… ≡ ¬□(¬φ ∨ □¬ψ)›
2715    using "KBasic:11" "≡E"(6) "oth-class-taut:3:a" by blast
2716  also AOT_have ‹… ≡ ¬(□¬φ ∨ □¬ψ)›
2717    using "S5Basic:9" "≡E"(1) "oth-class-taut:4:b" by blast
2718  also AOT_have ‹… ≡ ¬(¬◇φ ∨ ¬◇ψ)›
2719    apply (AOT_subst "«□¬φ»" "«¬◇φ»")
2720     apply (simp add: "KBasic2:1")
2721    apply (AOT_subst "«□¬ψ»" "«¬◇ψ»")
2722    by (auto simp: "KBasic2:1" "oth-class-taut:3:a")
2723  also AOT_have ‹… ≡ ◇φ & ◇ψ›
2724    using "≡E"(6) "oth-class-taut:3:a" "oth-class-taut:5:a" by blast
2725  finally show ?thesis .
2726qed
2727
2728AOT_theorem "S5Basic:12": ‹◇(φ & □ψ) ≡ (◇φ & □ψ)›
2729proof (rule "≡I"; rule "→I")
2730  AOT_assume ‹◇(φ & □ψ)›
2731  AOT_hence ‹◇φ & ◇□ψ›
2732    using "KBasic2:3" "vdash-properties:6" by blast
2733  AOT_thus ‹◇φ & □ψ›
2734    using "5◇" "&I" "&E"(1) "&E"(2) "vdash-properties:6" by blast
2735next
2736  AOT_assume ‹◇φ & □ψ›
2737  moreover AOT_have ‹(□□ψ & ◇φ) → ◇(φ & □ψ)›
2738    by (AOT_subst "«φ & □ψ»" "«□ψ & φ»")
2739       (auto simp: "Commutativity of &" "KBasic:16")
2740  ultimately AOT_show ‹◇(φ & □ψ)›
2741    by (metis "4" "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "vdash-properties:6")
2742qed
2743
2744
2745AOT_theorem "S5Basic:13": ‹□(φ → □ψ) ≡ □(◇φ → ψ)›
2746proof (rule "≡I")
2747  AOT_modally_strict {
2748    AOT_have ‹□(φ → □ψ) → (◇φ → ψ)›
2749      by (meson "KBasic:13" "B◇" "Hypothetical Syllogism" "deduction-theorem")
2750  }
2751  AOT_hence ‹□□(φ → □ψ) → □(◇φ → ψ)›
2752    by (rule RM)
2753  AOT_thus  ‹□(φ → □ψ) → □(◇φ → ψ)›
2754    using "4" "Hypothetical Syllogism" by blast
2755next
2756  AOT_modally_strict {
2757    AOT_have ‹□(◇φ → ψ) → (φ → □ψ)›
2758      by (meson "B" "Hypothetical Syllogism" "deduction-theorem" "qml:1" "vdash-properties:1[2]")
2759  }
2760  AOT_hence  ‹□□(◇φ → ψ) → □(φ → □ψ)›
2761    by (rule RM)
2762  AOT_thus ‹□(◇φ → ψ) → □(φ → □ψ)›
2763    using "4" "Hypothetical Syllogism" by blast
2764qed
2765
2766AOT_theorem "derived-S5-rules:1":
2767  assumes ‹Γ ⊢□ ◇φ → ψ› shows ‹□Γ ⊢□ φ → □ψ›
2768proof -
2769  AOT_have ‹□Γ ⊢□ □◇φ → □ψ›
2770    using assms by (rule "RM:1[prem]")
2771  AOT_thus ‹□Γ ⊢□ φ → □ψ›
2772    using "B" "Hypothetical Syllogism" by blast
2773qed
2774
2775AOT_theorem "derived-S5-rules:2":
2776  assumes ‹Γ ⊢□ φ → □ψ› shows ‹□Γ ⊢□ ◇φ → ψ›
2777proof -
2778  AOT_have ‹□Γ ⊢□ ◇φ → ◇□ψ›
2779    using assms by (rule "RM:2[prem]")
2780  AOT_thus ‹□Γ ⊢□ ◇φ → ψ›
2781    using "B◇" "Hypothetical Syllogism" by blast
2782qed
2783
2784AOT_theorem "BFs:1": ‹∀α □φ{α} → □∀α φ{α}›
2785proof -
2786  AOT_modally_strict {
2787    AOT_modally_strict {
2788      AOT_have ‹∀α □φ{α} → □φ{α}› for α by (fact AOT)
2789    }
2790    AOT_hence ‹◇∀α □φ{α} → ◇□φ{α}› for α by (rule "RM◇")
2791    AOT_hence ‹◇∀α □φ{α} → ∀α φ{α}›
2792      using "B◇" "∀I" "→E" "→I" by metis
2793  }
2794  thus ?thesis using "derived-S5-rules:1" by blast
2795qed
2796lemmas "BF" = "BFs:1"
2797
2798AOT_theorem "BFs:2": ‹□∀α φ{α} → ∀α □φ{α}›
2799proof -
2800  AOT_have ‹□∀α φ{α} → □φ{α}› for α using RM "cqt-orig:3" by metis
2801  thus ?thesis using  "cqt-orig:2"[THEN "→E"] "∀I" by metis
2802qed
2803lemmas "CBF" = "BFs:2"
2804
2805AOT_theorem "BFs:3": ‹◇∃α φ{α} → ∃α ◇φ{α}›
2806proof(rule "→I")
2807  AOT_modally_strict {
2808    AOT_have ‹□∀α ¬φ{α} ≡ ∀α □¬φ{α}›
2809      using BF CBF "≡I" by blast
2810  } note θ = this
2811
2812  AOT_assume ‹◇∃α φ{α}›
2813  AOT_hence ‹¬□¬(∃α φ{α})›
2814    using "≡dfE" "conventions:5" by blast
2815  AOT_hence ‹¬□∀α ¬φ{α}›
2816    apply - apply (AOT_subst "«∀α ¬φ{α}»" "«¬(∃α φ{α})»")
2817    using "≡dfI" "conventions:3" "conventions:4" "&I" "contraposition:2" "cqt-further:4"
2818          "df-rules-formulas[1]" "vdash-properties:1[2]" by blast
2819  AOT_hence ‹¬∀α □¬φ{α}›
2820    apply - apply (AOT_subst_using_rev subst: θ)
2821    using θ by blast
2822  AOT_hence ‹¬∀α ¬¬□¬φ{α}›
2823    apply - apply (AOT_subst_rev "λ τ. «□¬φ{τ}»"  "λ τ. «¬¬□¬φ{τ}»")
2824    by (simp add: "oth-class-taut:3:b")
2825  AOT_hence 0: ‹∃α ¬□¬φ{α}›
2826    by (rule "conventions:4"[THEN "≡dfI"])
2827  AOT_show ‹∃α ◇φ{α}›
2828    apply (AOT_subst "λ τ . «◇φ{τ}»" "λ τ . «¬□¬φ{τ}»")
2829     apply (simp add: "conventions:5" "≡Df")
2830    using 0 by blast
2831qed
2832lemmas "BF◇" = "BFs:3"
2833
2834AOT_theorem "BFs:4": ‹∃α ◇φ{α} → ◇∃α φ{α}›
2835proof(rule "→I")
2836  AOT_assume ‹∃α ◇φ{α}›
2837  AOT_hence ‹¬∀α ¬◇φ{α}›
2838    using "conventions:4"[THEN "≡dfE"] by blast
2839  AOT_hence ‹¬∀α □¬φ{α}›
2840    apply - apply (AOT_subst "λ τ . «□¬φ{τ}»" "λ τ . «¬◇φ{τ}»")
2841    by (simp add: "KBasic2:1")
2842  moreover AOT_have ‹∀α □¬φ{α} ≡ □∀α ¬φ{α}›
2843    using "≡I" "BF" "CBF" by metis
2844  ultimately AOT_have 1: ‹¬□∀α ¬φ{α}›
2845    using "≡E"(3) by blast
2846  AOT_show ‹◇∃α φ{α}›
2847    apply (rule "conventions:5"[THEN "≡dfI"])
2848    apply (AOT_subst "«∃α φ{α}»" "«¬∀α ¬φ{α}»")
2849     apply (simp add: "conventions:4" "≡Df")
2850    apply (AOT_subst "«¬¬∀α ¬φ{α}»" "«∀α ¬φ{α}»")
2851    by (auto simp: 1 "≡I" "useful-tautologies:1" "useful-tautologies:2")
2852qed
2853lemmas "CBF◇" = "BFs:4"
2854
2855AOT_theorem "sign-S5-thm:1": ‹∃α □φ{α} → □∃α φ{α}›
2856proof(rule "→I")
2857  AOT_assume ‹∃α □φ{α}›
2858  then AOT_obtain α where ‹□φ{α}› using "∃E" by metis
2859  moreover AOT_have ‹□α↓›
2860    by (simp add: "ex:1:a" "rule-ui:2[const_var]" RN)
2861  moreover AOT_have ‹□φ{τ}, □τ↓ ⊢□ □∃α φ{α}› for τ
2862  proof -
2863    AOT_have ‹φ{τ}, τ↓ ⊢□ ∃α φ{α}› using "existential:1" by blast
2864    AOT_thus ‹□φ{τ}, □τ↓ ⊢□ □∃α φ{α}›
2865      using "RN[prem]"[where Γ="{φ τ, «τ↓»}", simplified] by blast
2866  qed
2867  ultimately AOT_show ‹□∃α φ{α}› by blast
2868qed
2869lemmas Buridan = "sign-S5-thm:1"
2870
2871AOT_theorem "sign-S5-thm:2": ‹◇∀α φ{α} → ∀α ◇φ{α}›
2872proof -
2873  AOT_have ‹∀α (◇∀α φ{α} → ◇φ{α})›
2874    by (simp add: "RM◇" "cqt-orig:3" "∀I")
2875  AOT_thus ‹◇∀α φ{α} → ∀α ◇φ{α}›
2876    using "∀E"(4) "∀I" "→E" "→I" by metis
2877qed
2878lemmas "Buridan◇" = "sign-S5-thm:2"
2879
2880AOT_theorem "sign-S5-thm:3": ‹◇∃α (φ{α} & ψ{α}) → ◇(∃α φ{α} & ∃α ψ{α})›
2881  apply (rule "RM:2")
2882  by (metis (no_types, lifting) "instantiation" "&I" "&E"(1)
2883                                "&E"(2) "deduction-theorem" "existential:2[const_var]")
2884
2885AOT_theorem "sign-S5-thm:4": ‹◇∃α (φ{α} & ψ{α}) → ◇∃α φ{α}›
2886  apply (rule "RM:2")
2887  by (meson "instantiation" "&E"(1) "deduction-theorem" "existential:2[const_var]")
2888
2889AOT_theorem "sign-S5-thm:5": ‹(□∀α (φ{α} → ψ{α}) & □∀α (ψ{α} → χ{α})) → □∀α (φ{α} → χ{α})›
2890proof -
2891  {
2892    fix φ' ψ' χ'
2893    AOT_assume ‹⊢□ φ' & ψ' → χ'›
2894    AOT_hence ‹□φ' & □ψ' → □χ'›
2895      using "RN[prem]"[where Γ="{φ', ψ'}"] apply simp
2896      using "&E" "&I" "→E" "→I" by metis
2897  } note R = this
2898  show ?thesis by (rule R; fact AOT)
2899qed
2900
2901AOT_theorem "sign-S5-thm:6": ‹(□∀α (φ{α} ≡ ψ{α}) & □∀α(ψ{α} ≡ χ{α})) → □∀α(φ{α} ≡ χ{α})›
2902proof -
2903  {
2904    fix φ' ψ' χ'
2905    AOT_assume ‹⊢□ φ' & ψ' → χ'›
2906    AOT_hence ‹□φ' & □ψ' → □χ'›
2907      using "RN[prem]"[where Γ="{φ', ψ'}"] apply simp
2908      using "&E" "&I" "→E" "→I" by metis
2909  } note R = this
2910  show ?thesis by (rule R; fact AOT)
2911qed
2912
2913AOT_theorem "exist-nec2:1": ‹◇τ↓ → τ↓›
2914  using "B◇" "RM◇" "Hypothetical Syllogism" "exist-nec" by blast
2915
2916AOT_theorem "exists-nec2:2": ‹◇τ↓ ≡ □τ↓›
2917  by (meson "Act-Sub:3" "Hypothetical Syllogism" "exist-nec" "exist-nec2:1" "≡I" "nec-imp-act")
2918
2919AOT_theorem "exists-nec2:3": ‹¬τ↓ → □¬τ↓›
2920  using "KBasic2:1" "deduction-theorem" "exist-nec2:1" "≡E"(2) "modus-tollens:1" by blast
2921
2922AOT_theorem "exists-nec2:4": ‹◇¬τ↓ ≡ □¬τ↓›
2923  by (metis "Act-Sub:3" "KBasic:12" "deduction-theorem" "exist-nec" "exists-nec2:3" "≡I" "≡E"(4) "nec-imp-act" "reductio-aa:1")
2924
2925AOT_theorem "id-nec2:1": ‹◇α = β → α = β›
2926  using "B◇" "RM◇" "Hypothetical Syllogism" "id-nec:1" by blast
2927
2928AOT_theorem "id-nec2:2": ‹α ≠ β → □α ≠ β›
2929  apply (AOT_subst_using subst: "=-infix"[THEN "≡Df"])
2930  using "KBasic2:1" "deduction-theorem" "id-nec2:1" "≡E"(2) "modus-tollens:1" by blast
2931
2932AOT_theorem "id-nec2:3": ‹◇α ≠ β → α ≠ β›
2933  apply (AOT_subst_using subst: "=-infix"[THEN "≡Df"])
2934  by (metis "KBasic:11" "deduction-theorem" "id-nec:2" "≡E"(3) "reductio-aa:2" "vdash-properties:6")
2935
2936AOT_theorem "id-nec2:4": ‹◇α = β → □α = β›
2937  using "Hypothetical Syllogism" "id-nec2:1" "id-nec:1" by blast
2938
2939AOT_theorem "id-nec2:5": ‹◇α ≠ β → □α ≠ β›
2940  using "id-nec2:3" "id-nec2:2" "→I" "→E" by metis
2941
2942AOT_theorem "sc-eq-box-box:1": ‹□(φ → □φ) ≡ (◇φ → □φ)›
2943  apply (rule "≡I"; rule "→I")
2944  using "KBasic:13" "5◇" "Hypothetical Syllogism" "vdash-properties:10" apply blast
2945  by (metis "KBasic2:1" "KBasic:1" "KBasic:2" "S5Basic:13" "≡E"(2) "raa-cor:5" "vdash-properties:6")
2946
2947AOT_theorem "sc-eq-box-box:2": ‹(□(φ → □φ) ∨ (◇φ → □φ)) → (◇φ ≡ □φ)›
2948  by (metis "Act-Sub:3" "KBasic:13" "5◇" "∨E"(2) "deduction-theorem" "≡I" "nec-imp-act" "raa-cor:2" "vdash-properties:10")
2949
2950AOT_theorem "sc-eq-box-box:3": ‹□(φ → □φ) → (¬□φ ≡ □¬φ)›
2951proof (rule "→I"; rule "≡I"; rule "→I")
2952  AOT_assume ‹□(φ → □φ)›
2953  AOT_hence ‹◇φ → □φ› using "sc-eq-box-box:1" "≡E" by blast
2954  moreover AOT_assume ‹¬□φ›
2955  ultimately AOT_have ‹¬◇φ›
2956    using "modus-tollens:1" by blast
2957  AOT_thus ‹□¬φ›
2958    using "KBasic2:1" "≡E"(2) by blast
2959next
2960  AOT_assume ‹□(φ → □φ)›
2961  moreover AOT_assume ‹□¬φ›
2962  ultimately AOT_show ‹¬□φ›
2963    using "modus-tollens:1" "qml:2" "vdash-properties:10" "vdash-properties:1[2]" by blast
2964qed
2965
2966AOT_theorem "sc-eq-box-box:4": ‹(□(φ → □φ) & □(ψ → □ψ)) → ((□φ ≡ □ψ) → □(φ ≡ ψ))›
2967proof(rule "→I"; rule "→I")
2968  AOT_assume θ: ‹□(φ → □φ) & □(ψ → □ψ)›
2969  AOT_assume ξ: ‹□φ ≡ □ψ›
2970  AOT_hence ‹(□φ & □ψ) ∨ (¬□φ & ¬□ψ)›
2971    using "≡E"(4) "oth-class-taut:4:g" "raa-cor:3" by blast
2972  moreover {
2973    AOT_assume ‹□φ & □ψ›
2974    AOT_hence ‹□(φ ≡ ψ)›
2975      using "KBasic:3" "KBasic:8" "≡E"(2) "vdash-properties:10" by blast
2976  }
2977  moreover {
2978    AOT_assume ‹¬□φ & ¬□ψ›
2979    moreover AOT_have ‹¬□φ ≡ □¬φ› and ‹¬□ψ ≡ □¬ψ›
2980      using θ "Conjunction Simplification"(1) "Conjunction Simplification"(2) "sc-eq-box-box:3" "vdash-properties:10" by metis+
2981    ultimately AOT_have ‹□¬φ & □¬ψ›
2982      by (metis "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "≡E"(4) "modus-tollens:1" "raa-cor:3")
2983    AOT_hence ‹□(φ ≡ ψ)›
2984      using "KBasic:3" "KBasic:9" "≡E"(2) "vdash-properties:10" by blast
2985  }
2986  ultimately AOT_show ‹□(φ ≡ ψ)›
2987    using "∨E"(2) "reductio-aa:1" by blast
2988qed
2989
2990AOT_theorem "sc-eq-box-box:5": ‹(□(φ → □φ) & □(ψ → □ψ)) → ((φ ≡ ψ) → □(φ ≡ ψ))›
2991proof (rule "→I"; rule "→I")
2992  AOT_assume A: ‹(□(φ → □φ) & □(ψ → □ψ))›
2993  AOT_hence ‹φ → □φ› and ‹ψ → □ψ›
2994    using "&E" "qml:2"[axiom_inst] "→E" by blast+
2995  moreover AOT_assume ‹φ ≡ ψ›
2996  ultimately AOT_have ‹□φ ≡ □ψ›
2997    using "→E" "qml:2"[axiom_inst] "≡E" "≡I" by meson
2998  moreover AOT_have ‹(□φ ≡ □ψ) → □(φ ≡ ψ)›
2999    using A "sc-eq-box-box:4" "→E" by blast
3000  ultimately AOT_show ‹□(φ ≡ ψ)› using "→E" by blast
3001qed
3002
3003AOT_theorem "sc-eq-box-box:6": ‹□(φ → □φ) → ((φ → □ψ) → □(φ → ψ))›
3004proof (rule "→I"; rule "→I"; rule "raa-cor:1")
3005  AOT_assume ‹¬□(φ → ψ)›
3006  AOT_hence 1: ‹◇¬(φ → ψ)› by (metis "KBasic:11" "≡E"(1))
3007  AOT_have ‹◇(φ & ¬ψ)›
3008    apply (AOT_subst ‹«φ & ¬ψ»› ‹«¬(φ → ψ)»›)
3009     apply (meson "Commutativity of ≡" "≡E"(1) "oth-class-taut:1:b")
3010    by (fact 1)
3011  AOT_hence ‹◇φ› and 2: ‹◇¬ψ› using "KBasic2:3"[THEN "→E"] "&E" by blast+
3012  moreover AOT_assume ‹□(φ → □φ)›
3013  ultimately AOT_have ‹□φ› by (metis "≡E"(1) "sc-eq-box-box:1" "→E")
3014  AOT_hence φ using "qml:2"[axiom_inst, THEN "→E"] by blast
3015  moreover AOT_assume ‹φ → □ψ›
3016  ultimately AOT_have ‹□ψ› using "→E" by blast
3017  moreover AOT_have ‹¬□ψ› using 2 "KBasic:12" "¬¬I" "intro-elim:3:d" by blast
3018  ultimately AOT_show ‹□ψ & ¬□ψ› using "&I" by blast
3019qed
3020
3021AOT_theorem "sc-eq-box-box:7": ‹□(φ → □φ) → ((φ → 𝒜ψ) → 𝒜(φ → ψ))›
3022proof (rule "→I"; rule "→I"; rule "raa-cor:1")
3023  AOT_assume ‹¬𝒜(φ → ψ)›
3024  AOT_hence 1: ‹𝒜¬(φ → ψ)› by (metis "Act-Basic:1" "∨E"(2))
3025  AOT_have ‹𝒜(φ & ¬ψ)›
3026    apply (AOT_subst ‹«φ & ¬ψ»› ‹«¬(φ → ψ)»›)
3027     apply (meson "Commutativity of ≡" "≡E"(1) "oth-class-taut:1:b")
3028    by (fact 1)
3029  AOT_hence ‹𝒜φ› and 2: ‹𝒜¬ψ› using "Act-Basic:2"[THEN "≡E"(1)] "&E" by blast+
3030  AOT_hence ‹◇φ› by (metis "Act-Sub:3" "→E")
3031  moreover AOT_assume ‹□(φ → □φ)›
3032  ultimately AOT_have ‹□φ› by (metis "≡E"(1) "sc-eq-box-box:1" "→E")
3033  AOT_hence φ using "qml:2"[axiom_inst, THEN "→E"] by blast
3034  moreover AOT_assume ‹φ → 𝒜ψ›
3035  ultimately AOT_have ‹𝒜ψ› using "→E" by blast
3036  moreover AOT_have ‹¬𝒜ψ› using 2 by (meson "Act-Sub:1" "≡E"(4) "raa-cor:3")
3037  ultimately AOT_show ‹𝒜ψ & ¬𝒜ψ› using "&I" by blast
3038qed
3039
3040AOT_theorem "sc-eq-fur:1": ‹◇𝒜φ ≡ □𝒜φ›
3041  using "Act-Basic:6" "Act-Sub:4" "≡E"(6) by blast
3042
3043AOT_theorem "sc-eq-fur:2": ‹□(φ → □φ) → (𝒜φ ≡ φ)›
3044  by (metis "B◇" "Act-Sub:3" "KBasic:13" "T◇" "Hypothetical Syllogism" "deduction-theorem" "≡I" "nec-imp-act")
3045
3046AOT_theorem "sc-eq-fur:3": ‹□∀x (φ{x} → □φ{x}) → (∃!x φ{x} → ιx φ{x}↓)›
3047proof (rule "→I"; rule "→I")
3048  AOT_assume ‹□∀x (φ{x} → □φ{x})›
3049  AOT_hence A: ‹∀x □(φ{x} → □φ{x})› using CBF "→E" by blast
3050  AOT_assume ‹∃!x φ{x}›
3051  then AOT_obtain a where a_def: ‹φ{a} & ∀y (φ{y} → y = a)›
3052    using "∃E"[rotated 1, OF "uniqueness:1"[THEN "≡dfE"]] by blast
3053  moreover AOT_have ‹□φ{a}› using calculation A "∀E"(2) "qml:2"[axiom_inst] "→E" "&E"(1) by blast
3054  AOT_hence ‹𝒜φ{a}› using "nec-imp-act" "vdash-properties:6" by blast
3055  moreover AOT_have ‹∀y (𝒜φ{y} → y = a)›
3056  proof (rule "∀I"; rule "→I")
3057    fix b
3058    AOT_assume ‹𝒜φ{b}›
3059    AOT_hence ‹◇φ{b}›
3060      using "Act-Sub:3" "vdash-properties:6" by blast
3061    moreover {
3062      AOT_have ‹□(φ{b} → □φ{b})›
3063        using A "∀E"(2) by blast
3064      AOT_hence ‹◇φ{b} → □φ{b}›
3065        using "KBasic:13" "5◇" "Hypothetical Syllogism" "vdash-properties:6" by blast
3066    }
3067    ultimately AOT_have ‹□φ{b}› using "→E" by blast
3068    AOT_hence ‹φ{b}› using "qml:2"[axiom_inst] "→E" by blast
3069    AOT_thus ‹b = a›
3070      using a_def[THEN "&E"(2)] "∀E"(2) "→E" by blast
3071  qed
3072  ultimately AOT_have ‹𝒜φ{a} & ∀y (𝒜φ{y} → y = a)›
3073    using "&I" by blast
3074  AOT_hence ‹∃x (𝒜φ{x} & ∀y (𝒜φ{y} → y = x))› using "∃I" by fast
3075  AOT_hence ‹∃!x 𝒜φ{x}› using "uniqueness:1"[THEN "≡dfI"] by fast
3076  AOT_thus ‹ιx φ{x}↓›
3077    using "actual-desc:1"[THEN "≡E"(2)] by blast
3078qed
3079
3080AOT_theorem "sc-eq-fur:4": ‹□∀x (φ{x} → □φ{x}) → (x = ιx φ{x} ≡ (φ{x} & ∀z (φ{z} → z = x)))›
3081proof (rule "→I")
3082  AOT_assume ‹□∀x (φ{x} → □φ{x})›
3083  AOT_hence ‹∀x □(φ{x} → □φ{x})› using CBF "→E" by blast
3084  AOT_hence A: ‹𝒜φ{α} ≡ φ{α}› for α using "sc-eq-fur:2" "∀E" "→E" by fast
3085  AOT_show ‹x = ιx φ{x} ≡ (φ{x} & ∀z (φ{z} → z = x))›
3086  proof (rule "≡I"; rule "→I")
3087    AOT_assume ‹x = ιx φ{x}›
3088    AOT_hence B: ‹𝒜φ{x} & ∀z (𝒜φ{z} → z = x)›
3089      using "nec-hintikka-scheme"[THEN "≡E"(1)] by blast
3090    AOT_show ‹φ{x} & ∀z (φ{z} → z = x)›
3091    proof (rule "&I"; (rule "∀I"; rule "→I")?)
3092      AOT_show ‹φ{x}› using A B[THEN "&E"(1)] "≡E"(1) by blast
3093    next
3094      AOT_show ‹z = x› if ‹φ{z}› for z
3095        using that B[THEN "&E"(2)] "∀E"(2) "→E" A[THEN "≡E"(2)] by blast
3096    qed
3097  next
3098    AOT_assume B: ‹φ{x} & ∀z (φ{z} → z = x)›
3099    AOT_have ‹𝒜φ{x} & ∀z (𝒜φ{z} → z = x)›
3100    proof(rule "&I"; (rule "∀I"; rule "→I")?)
3101      AOT_show ‹𝒜φ{x}› using B[THEN "&E"(1)] A[THEN "≡E"(2)] by blast
3102    next
3103      AOT_show ‹b = x› if ‹𝒜φ{b}› for b
3104        using that A[THEN "≡E"(1)] B[THEN "&E"(2), THEN "∀E"(2), THEN "→E"] by blast
3105    qed
3106    AOT_thus ‹x = ιx φ{x}›
3107      using "nec-hintikka-scheme"[THEN "≡E"(2)] by blast
3108  qed
3109qed
3110
3111AOT_theorem "id-act:1": ‹α = β ≡ 𝒜α = β›
3112  by (meson "Act-Sub:3" "Hypothetical Syllogism" "id-nec2:1" "id-nec:2" "≡I" "nec-imp-act")
3113
3114AOT_theorem "id-act:2": ‹α ≠ β ≡ 𝒜α ≠ β›
3115proof (AOT_subst "«α ≠ β»" "«¬(α = β)»")
3116  AOT_modally_strict {
3117    AOT_show ‹α ≠ β ≡ ¬(α = β)›
3118      by (simp add: "=-infix" "≡Df")
3119  }
3120next
3121  AOT_show ‹¬(α = β) ≡ 𝒜¬(α = β)›
3122  proof (safe intro!: "≡I" "→I")
3123    AOT_assume ‹¬α = β›
3124    AOT_hence ‹¬𝒜α = β› using "id-act:1" "≡E"(3) by blast
3125    AOT_thus ‹𝒜¬α = β›
3126      using "¬¬E" "Act-Sub:1" "≡E"(3) by blast
3127  next
3128    AOT_assume ‹𝒜¬α = β›
3129    AOT_hence ‹¬𝒜α = β›
3130      using "¬¬I" "Act-Sub:1" "≡E"(4) by blast
3131    AOT_thus ‹¬α = β›
3132      using "id-act:1" "≡E"(4) by blast
3133  qed
3134qed
3135
3136AOT_theorem "A-Exists:1": ‹𝒜∃!α φ{α} ≡ ∃!α 𝒜φ{α}›
3137proof -
3138  AOT_have ‹𝒜∃!α φ{α} ≡ 𝒜∃α∀β (φ{β} ≡ β = α)›
3139    by (AOT_subst_using subst: "uniqueness:2")
3140       (simp add: "oth-class-taut:3:a")
3141  also AOT_have ‹… ≡ ∃α 𝒜∀β (φ{β} ≡ β = α)›
3142    by (simp add: "Act-Basic:10")
3143  also AOT_have ‹… ≡ ∃α∀β 𝒜(φ{β} ≡ β = α)›
3144    by (AOT_subst "λ τ . «𝒜∀β (φ{β} ≡ β = τ)»" "λ τ . «∀β 𝒜(φ{β} ≡ β = τ)»")
3145       (auto simp: "logic-actual-nec:3" "vdash-properties:1[2]" "oth-class-taut:3:a")
3146  also AOT_have ‹… ≡ ∃α∀β (𝒜φ{β} ≡ 𝒜β = α)›
3147    by (AOT_subst_rev "λ τ τ' . «𝒜(φ{τ'} ≡ τ' = τ)»" "λ τ τ'. «𝒜φ{τ'} ≡ 𝒜τ' = τ»")
3148       (auto simp: "Act-Basic:5" "cqt-further:7")
3149  also AOT_have ‹… ≡ ∃α∀β (𝒜φ{β} ≡ β = α)›
3150    apply (AOT_subst "λ τ τ' :: 'a . «𝒜τ' = τ»" "λ τ τ'. «τ' = τ»")
3151     apply (meson "id-act:1" "≡E"(6) "oth-class-taut:3:a")
3152    by (simp add: "cqt-further:7")
3153  also AOT_have ‹... ≡ ∃!α 𝒜φ{α}›
3154    using "uniqueness:2" "Commutativity of ≡"[THEN "≡E"(1)] by fast
3155  finally show ?thesis .
3156qed
3157
3158AOT_theorem "A-Exists:2": ‹ιx φ{x}↓ ≡ 𝒜∃!x φ{x}›
3159  by (AOT_subst_using subst: "A-Exists:1")
3160     (simp add: "actual-desc:1")
3161
3162AOT_theorem "id-act-desc:1": ‹ιx (x = y)↓›
3163proof(rule "existence:1"[THEN "≡dfI"]; rule "∃I")
3164  AOT_show ‹[λx E!x → E!x]ιx (x = y)›
3165  proof (rule "russell-axiom[exe,1].nec-russell-axiom"[THEN "≡E"(2)]; rule "∃I"; (rule "&I")+)
3166    AOT_show ‹𝒜y = y› by (simp add: "RA[2]" "id-eq:1")
3167  next
3168    AOT_show ‹∀z (𝒜z = y → z = y)›
3169      apply (rule "∀I")
3170      using "id-act:1"[THEN "≡E"(2)] "→I" by blast
3171  next
3172    AOT_show ‹[λx E!x → E!x]y›
3173    proof (rule "lambda-predicates:2"[axiom_inst, THEN "→E", THEN "≡E"(2)])
3174      AOT_show ‹[λx E!x → E!x]↓›
3175        by "cqt:2[lambda]"
3176    next
3177      AOT_show ‹E!y → E!y› 
3178        by (simp add: "if-p-then-p")
3179    qed
3180  qed
3181next
3182  AOT_show ‹[λx E!x → E!x]↓›
3183    by "cqt:2[lambda]"
3184qed
3185
3186AOT_theorem "id-act-desc:2": ‹y = ιx (x = y)›
3187  by (rule descriptions[axiom_inst, THEN "≡E"(2)]; rule "∀I"; rule "id-act:1"[symmetric])
3188
3189AOT_theorem "pre-en-eq:1[1]": ‹x1[F] → □x1[F]›
3190  by (simp add: encoding "vdash-properties:1[2]")
3191
3192AOT_theorem "pre-en-eq:1[2]": ‹x1x2[F] → □x1x2[F]›
3193proof (rule "→I")
3194  AOT_assume ‹x1x2[F]›
3195  AOT_hence ‹x1[λy [F]yx2]› and ‹x2[λy [F]x1y]›
3196    using "nary-encoding[2]"[axiom_inst, THEN "≡E"(1)] "&E" by blast+
3197  moreover AOT_have ‹[λy [F]yx2]↓› by "cqt:2[lambda]"
3198  moreover AOT_have ‹[λy [F]x1y]↓› by "cqt:2[lambda]"
3199  ultimately AOT_have ‹□x1[λy [F]yx2]› and ‹□x2[λy [F]x1y]›
3200    using encoding[axiom_inst, unvarify F] "→E" "&I" by blast+
3201  note A = this
3202  AOT_hence ‹□(x1[λy [F]yx2] & x2[λy [F]x1y])›
3203    using "KBasic:3"[THEN "≡E"(2)] "&I" by blast
3204  AOT_thus ‹□x1x2[F]›
3205    by (rule "nary-encoding[2]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3206qed
3207
3208AOT_theorem "pre-en-eq:1[3]": ‹x1x2x3[F] → □x1x2x3[F]›
3209proof (rule "→I")
3210  AOT_assume ‹x1x2x3[F]›
3211  AOT_hence ‹x1[λy [F]yx2x3]› and ‹x2[λy [F]x1yx3]› and ‹x3[λy [F]x1x2y]›
3212    using "nary-encoding[3]"[axiom_inst, THEN "≡E"(1)] "&E" by blast+
3213  moreover AOT_have ‹[λy [F]yx2x3]↓› by "cqt:2[lambda]"
3214  moreover AOT_have ‹[λy [F]x1yx3]↓› by "cqt:2[lambda]"
3215  moreover AOT_have ‹[λy [F]x1x2y]↓› by "cqt:2[lambda]"
3216  ultimately AOT_have ‹□x1[λy [F]yx2x3]› and ‹□x2[λy [F]x1yx3]› and ‹□x3[λy [F]x1x2y]›
3217    using encoding[axiom_inst, unvarify F] "→E" by blast+
3218  note A = this
3219  AOT_have B: ‹□(x1[λy [F]yx2x3] & x2[λy [F]x1yx3] & x3[λy [F]x1x2y])›
3220    by (rule "KBasic:3"[THEN "≡E"(2)] "&I" A)+
3221  AOT_thus ‹□x1x2x3[F]›
3222    by (rule "nary-encoding[3]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3223qed
3224
3225AOT_theorem "pre-en-eq:1[4]": ‹x1x2x3x4[F] → □x1x2x3x4[F]›
3226proof (rule "→I")
3227  AOT_assume ‹x1x2x3x4[F]›
3228  AOT_hence ‹x1[λy [F]yx2x3x4]› and ‹x2[λy [F]x1yx3x4]› and ‹x3[λy [F]x1x2yx4]› and  ‹x4[λy [F]x1x2x3y]›
3229    using "nary-encoding[4]"[axiom_inst, THEN "≡E"(1)] "&E" by metis+
3230  moreover AOT_have ‹[λy [F]yx2x3x4]↓› by "cqt:2[lambda]"
3231  moreover AOT_have ‹[λy [F]x1yx3x4]↓› by "cqt:2[lambda]"
3232  moreover AOT_have ‹[λy [F]x1x2yx4]↓› by "cqt:2[lambda]"
3233  moreover AOT_have ‹[λy [F]x1x2x3y]↓› by "cqt:2[lambda]"
3234  ultimately AOT_have ‹□x1[λy [F]yx2x3x4]› and ‹□x2[λy [F]x1yx3x4]› and ‹□x3[λy [F]x1x2yx4]› and ‹□x4[λy [F]x1x2x3y]›
3235    using "→E" encoding[axiom_inst, unvarify F] by blast+
3236  note A = this
3237  AOT_have B: ‹□(x1[λy [F]yx2x3x4] & x2[λy [F]x1yx3x4] & x3[λy [F]x1x2yx4] & x4[λy [F]x1x2x3y])›
3238    by (rule "KBasic:3"[THEN "≡E"(2)] "&I" A)+
3239  AOT_thus ‹□x1x2x3x4[F]›
3240    by (rule "nary-encoding[4]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3241qed
3242
3243AOT_theorem "pre-en-eq:2[1]": ‹¬x1[F] → □¬x1[F]›
3244proof (rule "→I"; rule "raa-cor:1")
3245  AOT_assume ‹¬□¬x1[F]›
3246  AOT_hence ‹◇x1[F]›
3247    by (rule "conventions:5"[THEN "≡dfI"])
3248  AOT_hence ‹x1[F]›
3249    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[1]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3250  moreover AOT_assume ‹¬x1[F]›
3251  ultimately AOT_show ‹x1[F] & ¬x1[F]› by (rule "&I")
3252qed
3253AOT_theorem "pre-en-eq:2[2]": ‹¬x1x2[F] → □¬x1x2[F]›
3254proof (rule "→I"; rule "raa-cor:1")
3255  AOT_assume ‹¬□¬x1x2[F]›
3256  AOT_hence ‹◇x1x2[F]›
3257    by (rule "conventions:5"[THEN "≡dfI"])
3258  AOT_hence ‹x1x2[F]›
3259    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[2]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3260  moreover AOT_assume ‹¬x1x2[F]›
3261  ultimately AOT_show ‹x1x2[F] & ¬x1x2[F]› by (rule "&I")
3262qed
3263
3264AOT_theorem "pre-en-eq:2[3]": ‹¬x1x2x3[F] → □¬x1x2x3[F]›
3265proof (rule "→I"; rule "raa-cor:1")
3266  AOT_assume ‹¬□¬x1x2x3[F]›
3267  AOT_hence ‹◇x1x2x3[F]›
3268    by (rule "conventions:5"[THEN "≡dfI"])
3269  AOT_hence ‹x1x2x3[F]›
3270    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[3]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3271  moreover AOT_assume ‹¬x1x2x3[F]›
3272  ultimately AOT_show ‹x1x2x3[F] & ¬x1x2x3[F]› by (rule "&I")
3273qed
3274
3275AOT_theorem "pre-en-eq:2[4]": ‹¬x1x2x3x4[F] → □¬x1x2x3x4[F]›
3276proof (rule "→I"; rule "raa-cor:1")
3277  AOT_assume ‹¬□¬x1x2x3x4[F]›
3278  AOT_hence ‹◇x1x2x3x4[F]›
3279    by (rule "conventions:5"[THEN "≡dfI"])
3280  AOT_hence ‹x1x2x3x4[F]›
3281    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[4]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3282  moreover AOT_assume ‹¬x1x2x3x4[F]›
3283  ultimately AOT_show ‹x1x2x3x4[F] & ¬x1x2x3x4[F]› by (rule "&I")
3284qed
3285
3286AOT_theorem "en-eq:1[1]": ‹◇x1[F] ≡ □x1[F]›
3287  using "pre-en-eq:1[1]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by metis
3288AOT_theorem "en-eq:1[2]": ‹◇x1x2[F] ≡ □x1x2[F]›
3289  using "pre-en-eq:1[2]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by metis
3290AOT_theorem "en-eq:1[3]": ‹◇x1x2x3[F] ≡ □x1x2x3[F]›
3291  using "pre-en-eq:1[3]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by fast
3292AOT_theorem "en-eq:1[4]": ‹◇x1x2x3x4[F] ≡ □x1x2x3x4[F]›
3293  using "pre-en-eq:1[4]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by fast
3294
3295AOT_theorem "en-eq:2[1]": ‹x1[F] ≡ □x1[F]›
3296  by (simp add: "≡I" "pre-en-eq:1[1]" "qml:2"[axiom_inst])
3297AOT_theorem "en-eq:2[2]": ‹x1x2[F] ≡ □x1x2[F]›
3298  by (simp add: "≡I" "pre-en-eq:1[2]" "qml:2"[axiom_inst])
3299AOT_theorem "en-eq:2[3]": ‹x1x2x3[F] ≡ □x1x2x3[F]›
3300  by (simp add: "≡I" "pre-en-eq:1[3]" "qml:2"[axiom_inst])
3301AOT_theorem "en-eq:2[4]": ‹x1x2x3x4[F] ≡ □x1x2x3x4[F]›
3302  by (simp add: "≡I" "pre-en-eq:1[4]" "qml:2"[axiom_inst])
3303
3304AOT_theorem "en-eq:3[1]": ‹◇x1[F] ≡ x1[F]›
3305  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[1]"] "≡I" by blast
3306AOT_theorem "en-eq:3[2]": ‹◇x1x2[F] ≡ x1x2[F]›
3307  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[2]"] "≡I" by blast
3308AOT_theorem "en-eq:3[3]": ‹◇x1x2x3[F] ≡ x1x2x3[F]›
3309  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[3]"] "≡I" by blast
3310AOT_theorem "en-eq:3[4]": ‹◇x1x2x3x4[F] ≡ x1x2x3x4[F]›
3311  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[4]"] "≡I" by blast
3312
3313AOT_theorem "en-eq:4[1]": ‹(x1[F] ≡ y1[G]) ≡ (□x1[F] ≡ □y1[G])›
3314  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3315  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[1]" by blast+
3316AOT_theorem "en-eq:4[2]": ‹(x1x2[F] ≡ y1y2[G]) ≡ (□x1x2[F] ≡ □y1y2[G])›
3317  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3318  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[2]" by blast+
3319AOT_theorem "en-eq:4[3]": ‹(x1x2x3[F] ≡ y1y2y3[G]) ≡ (□x1x2x3[F] ≡ □y1y2y3[G])›
3320  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3321  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[3]" by blast+
3322AOT_theorem "en-eq:4[4]": ‹(x1x2x3x4[F] ≡ y1y2y3y4[G]) ≡ (□x1x2x3x4[F] ≡ □y1y2y3y4[G])›
3323  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3324  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[4]" by blast+
3325
3326AOT_theorem "en-eq:5[1]": ‹□(x1[F] ≡ y1[G]) ≡ (□x1[F] ≡ □y1[G])›
3327  apply (rule "≡I"; rule "→I")
3328  using "en-eq:4[1]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3329  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3330        "&I"[OF "pre-en-eq:1[1]"[THEN RN], OF "pre-en-eq:1[1]"[THEN RN]] by blast
3331AOT_theorem "en-eq:5[2]": ‹□(x1x2[F] ≡ y1y2[G]) ≡ (□x1x2[F] ≡ □y1y2[G])›
3332  apply (rule "≡I"; rule "→I")
3333  using "en-eq:4[2]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3334  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3335        "&I"[OF "pre-en-eq:1[2]"[THEN RN], OF "pre-en-eq:1[2]"[THEN RN]] by blast
3336AOT_theorem "en-eq:5[3]": ‹□(x1x2x3[F] ≡ y1y2y3[G]) ≡ (□x1x2x3[F] ≡ □y1y2y3[G])›
3337  apply (rule "≡I"; rule "→I")
3338  using "en-eq:4[3]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3339  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3340        "&I"[OF "pre-en-eq:1[3]"[THEN RN], OF "pre-en-eq:1[3]"[THEN RN]] by blast
3341AOT_theorem "en-eq:5[4]": ‹□(x1x2x3x4[F] ≡ y1y2y3y4[G]) ≡ (□x1x2x3x4[F] ≡ □y1y2y3y4[G])›
3342  apply (rule "≡I"; rule "→I")
3343  using "en-eq:4[4]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3344  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3345        "&I"[OF "pre-en-eq:1[4]"[THEN RN], OF "pre-en-eq:1[4]"[THEN RN]] by blast
3346
3347AOT_theorem "en-eq:6[1]": ‹(x1[F] ≡ y1[G]) ≡ □(x1[F] ≡ y1[G])›
3348  using "en-eq:5[1]"[symmetric] "en-eq:4[1]" "≡E"(5) by fast
3349AOT_theorem "en-eq:6[2]": ‹(x1x2[F] ≡ y1y2[G]) ≡ □(x1x2[F] ≡ y1y2[G])›
3350  using "en-eq:5[2]"[symmetric] "en-eq:4[2]" "≡E"(5) by fast
3351AOT_theorem "en-eq:6[3]": ‹(x1x2x3[F] ≡ y1y2y3[G]) ≡ □(x1x2x3[F] ≡ y1y2y3[G])›
3352  using "en-eq:5[3]"[symmetric] "en-eq:4[3]" "≡E"(5) by fast
3353AOT_theorem "en-eq:6[4]": ‹(x1x2x3x4[F] ≡ y1y2y3y4[G]) ≡ □(x1x2x3x4[F] ≡ y1y2y3y4[G])›
3354  using "en-eq:5[4]"[symmetric] "en-eq:4[4]" "≡E"(5) by fast
3355
3356AOT_theorem "en-eq:7[1]": ‹¬x1[F] ≡ □¬x1[F]›
3357  using "pre-en-eq:2[1]" "qml:2"[axiom_inst] "≡I" by blast
3358AOT_theorem "en-eq:7[2]": ‹¬x1x2[F] ≡ □¬x1x2[F]›
3359  using "pre-en-eq:2[2]" "qml:2"[axiom_inst] "≡I" by blast
3360AOT_theorem "en-eq:7[3]": ‹¬x1x2x3[F] ≡ □¬x1x2x3[F]›
3361  using "pre-en-eq:2[3]" "qml:2"[axiom_inst] "≡I" by blast
3362AOT_theorem "en-eq:7[4]": ‹¬x1x2x3x4[F] ≡ □¬x1x2x3x4[F]›
3363  using "pre-en-eq:2[4]" "qml:2"[axiom_inst] "≡I" by blast
3364
3365AOT_theorem "en-eq:8[1]": ‹◇¬x1[F] ≡ ¬x1[F]›
3366  using "en-eq:2[1]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3367AOT_theorem "en-eq:8[2]": ‹◇¬x1x2[F] ≡ ¬x1x2[F]›
3368  using "en-eq:2[2]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3369AOT_theorem "en-eq:8[3]": ‹◇¬x1x2x3[F] ≡ ¬x1x2x3[F]›
3370  using "en-eq:2[3]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3371AOT_theorem "en-eq:8[4]": ‹◇¬x1x2x3x4[F] ≡ ¬x1x2x3x4[F]›
3372  using "en-eq:2[4]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3373
3374AOT_theorem "en-eq:9[1]": ‹◇¬x1[F] ≡ □¬x1[F]›
3375  using "en-eq:7[1]" "en-eq:8[1]" "≡E"(5) by blast
3376AOT_theorem "en-eq:9[2]": ‹◇¬x1x2[F] ≡ □¬x1x2[F]›
3377  using "en-eq:7[2]" "en-eq:8[2]" "≡E"(5) by blast
3378AOT_theorem "en-eq:9[3]": ‹◇¬x1x2x3[F] ≡ □¬x1x2x3[F]›
3379  using "en-eq:7[3]" "en-eq:8[3]" "≡E"(5) by blast
3380AOT_theorem "en-eq:9[4]": ‹◇¬x1x2x3x4[F] ≡ □¬x1x2x3x4[F]›
3381  using "en-eq:7[4]" "en-eq:8[4]" "≡E"(5) by blast
3382
3383AOT_theorem "en-eq:10[1]": ‹𝒜x1[F] ≡ x1[F]›
3384  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[1]" "pre-en-eq:1[1]")
3385AOT_theorem "en-eq:10[2]": ‹𝒜x1x2[F] ≡ x1x2[F]›
3386  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[2]" "pre-en-eq:1[2]")
3387AOT_theorem "en-eq:10[3]": ‹𝒜x1x2x3[F] ≡ x1x2x3[F]›
3388  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[3]" "pre-en-eq:1[3]")
3389AOT_theorem "en-eq:10[4]": ‹𝒜x1x2x3x4[F] ≡ x1x2x3x4[F]›
3390  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[4]" "pre-en-eq:1[4]")
3391
3392AOT_theorem "oa-facts:1": ‹O!x → □O!x›
3393proof(rule "→I")
3394  AOT_modally_strict {
3395    AOT_have ‹[λx ◇E!x]x ≡ ◇E!x›
3396      by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
3397  } note θ = this
3398  AOT_assume ‹O!x›
3399  AOT_hence ‹[λx ◇E!x]x›
3400    by (rule "=dfE"(2)[OF AOT_ordinary, rotated 1]) "cqt:2[lambda]"
3401  AOT_hence ‹◇E!x› using θ[THEN "≡E"(1)] by blast
3402  AOT_hence 0: ‹□◇E!x› using "qml:3"[axiom_inst, THEN "→E"] by blast
3403  AOT_have ‹□[λx ◇E!x]x›
3404    by (AOT_subst_using subst: θ) (simp add: 0)
3405  AOT_thus ‹□O!x›
3406    by (rule "=dfI"(2)[OF AOT_ordinary, rotated 1]) "cqt:2[lambda]"
3407qed
3408
3409AOT_theorem "oa-facts:2": ‹A!x → □A!x›
3410proof(rule "→I")
3411  AOT_modally_strict {
3412    AOT_have ‹[λx ¬◇E!x]x ≡ ¬◇E!x›
3413      by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
3414  } note θ = this
3415  AOT_assume ‹A!x›
3416  AOT_hence ‹[λx ¬◇E!x]x›
3417    by (rule "=dfE"(2)[OF AOT_abstract, rotated 1]) "cqt:2[lambda]"
3418  AOT_hence ‹¬◇E!x› using θ[THEN "≡E"(1)] by blast
3419  AOT_hence ‹□¬E!x› using "KBasic2:1"[THEN "≡E"(2)] by blast
3420  AOT_hence 0: ‹□□¬E!x› using "4"[THEN "→E"] by blast
3421  AOT_have 1: ‹□¬◇E!x›
3422    apply (AOT_subst "«¬◇E!x»" "«□¬E!x»")
3423    using "KBasic2:1"[symmetric] apply blast
3424    using 0 by blast
3425  AOT_have ‹□[λx ¬◇E!x]x›
3426    by (AOT_subst_using subst: θ) (simp add: 1)
3427  AOT_thus ‹□A!x›
3428    by (rule "=dfI"(2)[OF AOT_abstract, rotated 1]) "cqt:2[lambda]"
3429qed
3430
3431AOT_theorem "oa-facts:3": ‹◇O!x → O!x›
3432  using "oa-facts:1" "B◇" "RM◇" "Hypothetical Syllogism" by blast
3433AOT_theorem "oa-facts:4": ‹◇A!x → A!x›
3434  using "oa-facts:2" "B◇" "RM◇" "Hypothetical Syllogism" by blast
3435
3436AOT_theorem "oa-facts:5": ‹◇O!x ≡ □O!x›
3437  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:1" "oa-facts:3")
3438
3439AOT_theorem "oa-facts:6": ‹◇A!x ≡ □A!x›
3440  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:2" "oa-facts:4")
3441
3442AOT_theorem "oa-facts:7": ‹O!x ≡ 𝒜O!x›
3443  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:1" "oa-facts:3")
3444
3445AOT_theorem "oa-facts:8": ‹A!x ≡ 𝒜A!x›
3446  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:2" "oa-facts:4")
3447
3448AOT_theorem "beta-C-meta": ‹[λμ1...μn φ{μ1...μn, ν1...νn}]↓ → ([λμ1...μn φ{μ1...μn, ν1...νn}]ν1...νn ≡ φ{ν1...νn, ν1...νn})›
3449  using "lambda-predicates:2"[axiom_inst] by blast
3450
3451AOT_theorem "beta-C-cor:1": ‹(∀ν1...∀νn([λμ1...μn φ{μ1...μn, ν1...νn}]↓)) → ∀ν1...∀νn ([λμ1...μn φ{μ1...μn, ν1...νn}]ν1...νn ≡ φ{ν1...νn, ν1...νn})›
3452  apply (rule "cqt-basic:14"[where 'a='a, THEN "→E"])
3453  using "beta-C-meta" "∀I" by fast
3454
3455AOT_theorem "beta-C-cor:2": ‹[λμ1...μn φ{μ1...μn}]↓ → ∀ν1...∀νn ([λμ1...μn φ{μ1...μn}]ν1...νn ≡ φ{ν1...νn})›
3456  apply (rule "→I"; rule "∀I")
3457  using "beta-C-meta"[THEN "→E"] by fast
3458
3459(* TODO: syntax + double-check if this is really a faithful representation *)
3460theorem "beta-C-cor:3": assumes ‹⋀ν1νn. AOT_instance_of_cqt_2 (φ (AOT_term_of_var ν1νn))›
3461  shows ‹[v ⊨ ∀ν1...∀νn ([λμ1...μn φ{ν1...νn,μ1...μn}]ν1...νn ≡ φ{ν1...νn,ν1...νn})]›
3462  using "cqt:2[lambda]"[axiom_inst, OF assms] "beta-C-cor:1"[THEN "→E"] "∀I" by fast
3463
3464AOT_theorem "betaC:1:a": ‹[λμ1...μn φ{μ1...μn}]κ1...κn ⊢□ φ{κ1...κn}›
3465proof -
3466  AOT_modally_strict {
3467    AOT_assume ‹[λμ1...μn φ{μ1...μn}]κ1...κn›
3468    moreover AOT_have ‹[λμ1...μn φ{μ1...μn}]↓› and ‹κ1...κn↓›
3469      using calculation "cqt:5:a"[axiom_inst, THEN "→E"] "&E" by blast+
3470    ultimately AOT_show ‹φ{κ1...κn}›
3471      using "beta-C-cor:2"[THEN "→E", THEN "∀E"(1), THEN "≡E"(1)] by blast
3472  }
3473qed
3474
3475AOT_theorem "betaC:1:b": ‹¬φ{κ1...κn} ⊢□ ¬[λμ1...μn φ{μ1...μn}]κ1...κn›
3476  using "betaC:1:a" "raa-cor:3" by blast
3477
3478lemmas "β→C" = "betaC:1:a" "betaC:1:b"
3479
3480AOT_theorem "betaC:2:a": ‹[λμ1...μn φ{μ1...μn}]↓, κ1...κn↓, φ{κ1...κn} ⊢□ [λμ1...μn φ{μ1...μn}]κ1...κn›
3481proof -
3482  AOT_modally_strict {
3483    AOT_assume 1: ‹[λμ1...μn φ{μ1...μn}]↓› and 2: ‹κ1...κn↓› and 3: ‹φ{κ1...κn}›
3484    AOT_hence ‹[λμ1...μn φ{μ1...μn}]κ1...κn›
3485      using "beta-C-cor:2"[THEN "→E", OF 1, THEN "∀E"(1), THEN "≡E"(2)] by blast
3486  }
3487  AOT_thus ‹[λμ1...μn φ{μ1...μn}]↓, κ1...κn↓, φ{κ1...κn} ⊢□ [λμ1...μn φ{μ1...μn}]κ1...κn›
3488    by blast
3489qed
3490
3491AOT_theorem "betaC:2:b": ‹[λμ1...μn φ{μ1...μn}]↓, κ1...κn↓, ¬[λμ1...μn φ{μ1...μn}]κ1...κn ⊢□ ¬φ{κ1...κn}›
3492  using "betaC:2:a" "raa-cor:3" by blast
3493
3494lemmas "β←C" = "betaC:2:a" "betaC:2:b"
3495
3496AOT_theorem "eta-conversion-lemma1:1": ‹Π↓ → [λx1...xn [Π]x1...xn] = Π›
3497  using "lambda-predicates:3"[axiom_inst] "∀I" "∀E"(1) "→I" by fast
3498
3499AOT_theorem "eta-conversion-lemma1:2": ‹Π↓ → [λν1...νn [Π]ν1...νn] = Π›
3500  using "eta-conversion-lemma1:1". (* TODO: spurious in the embedding *)
3501
3502(* match (τ) in "λa . ?b" ⇒ ‹match (τ') in "λa . ?b" ⇒ ‹fail›› ¦ _ ⇒ ‹ *)
3503
3504text‹Note: not explicitly part of PLM.›
3505AOT_theorem id_sym: assumes ‹τ = τ'› shows ‹τ' = τ›
3506  using "rule=E"[where φ="λ τ' . «τ' = τ»", rotated 1, OF assms]
3507        "=I"(1)[OF "t=t-proper:1"[THEN "→E", OF assms]] by auto
3508declare id_sym[sym]
3509
3510text‹Note: not explicitly part of PLM.›
3511AOT_theorem id_trans: assumes ‹τ = τ'› and ‹τ' = τ''› shows ‹τ = τ''›
3512  using "rule=E" assms by blast
3513declare id_trans[trans]
3514
3515method "ηC" for Π :: ‹<'a::{AOT_Term_id_2,AOT_κs}>› = (match conclusion in "[v ⊨ τ{Π} = τ'{Π}]" for v τ τ' ⇒ ‹
3516rule "rule=E"[rotated 1, OF "eta-conversion-lemma1:2"[THEN "→E", of v "«[Π]»", symmetric]]
3517›)
3518(*
3519AOT_theorem ‹[λy [λz [P]z]y → [λu [S]u]y] = [λy [P]y → [S]y]›
3520  apply ("ηC" "«[P]»") defer
3521   apply ("ηC" "«[S]»") defer
3522  oops
3523*)
3524(* TODO: proper representation of eta_conversion_lemma2 *)
3525
3526AOT_theorem "sub-des-lam:1": ‹[λz1...zn  χ{z1...zn, ιx φ{x}}]↓ & ιx φ{x} = ιx ψ{x} → [λz1...zn χ{z1...zn, ιx φ{x}}] = [λz1...zn χ{z1...zn, ιx ψ{x}}]›
3527proof(rule "→I")
3528  AOT_assume A: ‹[λz1...zn  χ{z1...zn, ιx φ{x}}]↓ & ιx φ{x} = ιx ψ{x}›
3529  AOT_show ‹[λz1...zn χ{z1...zn, ιx φ{x}}] = [λz1...zn χ{z1...zn, ιx ψ{x}}]›
3530    using "rule=E"[where φ="λ τ . «[λz1...zn χ{z1...zn, ιx φ{x}}] = [λz1...zn χ{z1...zn, τ}]»",
3531               OF "=I"(1)[OF A[THEN "&E"(1)]], OF A[THEN "&E"(2)]]
3532    by blast
3533qed
3534
3535AOT_theorem "sub-des-lam:2": ‹ιx φ{x} = ιx ψ{x} → χ{ιx φ{x}} = χ{ιx ψ{x}}› for χ :: ‹κ ⇒ 𝗈›
3536  using "rule=E"[where φ="λ τ . «χ{ιx φ{x}} = χ{τ}»", OF "=I"(1)[OF "log-prop-prop:2"]] "→I" by blast
3537
3538AOT_theorem "prop-equiv": ‹F = G ≡ ∀x (x[F] ≡ x[G])›
3539proof(rule "≡I"; rule "→I")
3540  AOT_assume ‹F = G›
3541  AOT_thus ‹∀x (x[F] ≡ x[G])›
3542    by (rule "rule=E"[rotated]) (fact "oth-class-taut:3:a"[THEN GEN])
3543next
3544  AOT_assume ‹∀x (x[F] ≡ x[G])›
3545  AOT_hence ‹x[F] ≡ x[G]› for x using "∀E" by blast
3546  AOT_hence ‹□(x[F] ≡ x[G])› for x using "en-eq:6[1]"[THEN "≡E"(1)] by blast
3547  AOT_hence ‹∀x □(x[F] ≡ x[G])› by (rule GEN)
3548  AOT_hence ‹□∀x (x[F] ≡ x[G])› using BF[THEN "→E"] by fast
3549  AOT_thus "F = G" using "p-identity-thm2:1"[THEN "≡E"(2)] by blast
3550qed
3551
3552AOT_theorem "relations:1":
3553  assumes ‹INSTANCE_OF_CQT_2(φ)›
3554  shows ‹∃F □∀x1...∀xn ([F]x1...xn ≡ φ{x1...xn})›
3555  apply (rule "∃I"(1)[where τ="«[λx1...xn φ{x1...xn}]»"])
3556  using "cqt:2[lambda]"[OF assms, axiom_inst] "beta-C-cor:2"[THEN "→E", THEN RN] by blast+
3557
3558AOT_theorem "relations:2":
3559  assumes ‹INSTANCE_OF_CQT_2(φ)›
3560  shows ‹∃F □∀x ([F]x ≡ φ{x})›
3561  using "relations:1" assms by blast
3562
3563AOT_theorem "block-paradox:1": ‹¬[λx ∃G (x[G] & ¬[G]x)]↓›
3564proof(rule RAA(2))
3565  let ?φ="λ τ. «∃G (τ[G] & ¬[G]τ)»"
3566  AOT_assume A: ‹[λx «?φ x»]↓›
3567  AOT_have ‹∃x (A!x & ∀F (x[F] ≡ F = [λx «?φ x»]))›
3568    using "A-objects"[axiom_inst] by fast
3569  then AOT_obtain a where ξ: ‹A!a & ∀F (a[F] ≡ F = [λx «?φ x»])›
3570    using "∃E"[rotated] by blast
3571  AOT_show ‹¬[λx ∃G (x[G] & ¬[G]x)]↓›
3572  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3573    AOT_assume B: ‹[λx «?φ x»]a›
3574    AOT_hence ‹∃G (a[G] & ¬[G]a)› using "β→C" A by blast
3575    then AOT_obtain P where ‹a[P] & ¬[P]a› using "∃E"[rotated] by blast
3576    moreover AOT_have ‹P = [λx «?φ x»]›
3577      using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1)] calculation[THEN "&E"(1)] by blast
3578    ultimately AOT_have ‹¬[λx «?φ x»]a›
3579      using "rule=E" "&E"(2) by fast
3580    AOT_thus ‹¬[λx ∃G (x[G] & ¬[G]x)]↓› using B RAA by blast
3581  next
3582    AOT_assume B: ‹¬[λx «?φ x»]a›
3583    AOT_hence ‹¬∃G (a[G] & ¬[G]a)› using "β←C" "cqt:2[const_var]"[of a, axiom_inst] A by blast
3584    AOT_hence C: ‹∀G ¬(a[G] & ¬[G]a)› using "cqt-further:4"[THEN "→E"] by blast
3585    AOT_have ‹∀G (a[G] → [G]a)›
3586      by (AOT_subst "λ Π . «a[Π] → [Π]a»" "λ Π . «¬(a[Π] & ¬[Π]a)»")
3587         (auto simp: "oth-class-taut:1:a" C)
3588    AOT_hence ‹a[λx «?φ x»] → [λx «?φ x»]a› using "∀E" A by blast
3589    moreover AOT_have ‹a[λx «?φ x»]› using ξ[THEN "&E"(2), THEN "∀E"(1), OF A, THEN "≡E"(2)]
3590      using "=I"(1)[OF A] by blast
3591    ultimately AOT_show ‹¬[λx ∃G (x[G] & ¬[G]x)]↓› using B "→E" RAA by blast
3592  qed
3593qed(simp)
3594
3595AOT_theorem "block-paradox:2": ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))›
3596proof(rule RAA(2))
3597  AOT_assume ‹∃F ∀x ([F]x ≡ ∃G (x[G] & ¬[G]x))›
3598  then AOT_obtain F where F_prop: ‹∀x ([F]x ≡ ∃G (x[G] & ¬[G]x))› using "∃E"[rotated] by blast
3599  AOT_have ‹∃x (A!x & ∀G (x[G] ≡ G = F))›
3600    using "A-objects"[axiom_inst] by fast
3601  then AOT_obtain a where ξ: ‹A!a & ∀G (a[G] ≡ G = F)›
3602    using "∃E"[rotated] by blast
3603  AOT_show ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))›
3604  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3605    AOT_assume B: ‹[F]a›
3606    AOT_hence ‹∃G (a[G] & ¬[G]a)› using F_prop[THEN "∀E"(2), THEN "≡E"(1)] by blast
3607    then AOT_obtain P where ‹a[P] & ¬[P]a› using "∃E"[rotated] by blast
3608    moreover AOT_have ‹P = F›
3609      using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1)] calculation[THEN "&E"(1)] by blast
3610    ultimately AOT_have ‹¬[F]a›
3611      using "rule=E" "&E"(2) by fast
3612    AOT_thus ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))› using B RAA by blast
3613  next
3614    AOT_assume B: ‹¬[F]a›
3615    AOT_hence ‹¬∃G (a[G] & ¬[G]a)›
3616      using "oth-class-taut:4:b"[THEN "≡E"(1), OF F_prop[THEN "∀E"(2)[of _ _ a]], THEN "≡E"(1)] by simp
3617    AOT_hence C: ‹∀G ¬(a[G] & ¬[G]a)› using "cqt-further:4"[THEN "→E"] by blast
3618    AOT_have ‹∀G (a[G] → [G]a)›
3619      by (AOT_subst "λ Π . «a[Π] → [Π]a»" "λ Π . «¬(a[Π] & ¬[Π]a)»")
3620         (auto simp: "oth-class-taut:1:a" C)
3621    AOT_hence ‹a[F] → [F]a› using "∀E" by blast
3622    moreover AOT_have ‹a[F]› using ξ[THEN "&E"(2), THEN "∀E"(2), of F, THEN "≡E"(2)]
3623      using "=I"(2) by blast
3624    ultimately AOT_show ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))› using B "→E" RAA by blast
3625  qed
3626qed(simp)
3627
3628AOT_theorem "block-paradox:3": ‹¬∀y [λz z = y]↓›
3629proof(rule RAA(2))
3630  AOT_assume θ: ‹∀y [λz z = y]↓›
3631  AOT_have ‹∃x (A!x & ∀F (x[F] ≡ ∃y(F = [λz z = y] & ¬y[F])))›
3632    using "A-objects"[axiom_inst] by force
3633  then AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y (F = [λz z = y] & ¬y[F]))›
3634    using "∃E"[rotated] by blast
3635  AOT_have ζ: ‹a[λz z = a] ≡ ∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3636    using θ[THEN "∀E"(2)] a_prop[THEN "&E"(2), THEN "∀E"(1)] by blast
3637  AOT_show ‹¬∀y [λz z = y]↓›
3638  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3639    AOT_assume A: ‹a[λz z = a]›
3640    AOT_hence ‹∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3641      using ζ[THEN "≡E"(1)] by blast
3642    then AOT_obtain b where b_prop: ‹[λz z = a] = [λz z = b] & ¬b[λz z = a]›
3643      using "∃E"[rotated] by blast
3644    moreover AOT_have ‹a = a› by (rule "=I")
3645    moreover AOT_have ‹[λz z = a]↓› using θ "∀E" by blast
3646    moreover AOT_have ‹a↓› using "cqt:2[const_var]"[axiom_inst] .
3647    ultimately AOT_have ‹[λz z = a]a› using "β←C" by blast
3648    AOT_hence ‹[λz z = b]a› using "rule=E" b_prop[THEN "&E"(1)] by fast
3649    AOT_hence ‹a = b› using "β→C" by blast
3650    AOT_hence ‹b[λz z = a]› using A "rule=E" by fast
3651    AOT_thus ‹¬∀y [λz z = y]↓› using b_prop[THEN "&E"(2)] RAA by blast
3652  next
3653    AOT_assume A: ‹¬a[λz z = a]›
3654    AOT_hence ‹¬∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3655      using ζ "oth-class-taut:4:b"[THEN "≡E"(1), THEN "≡E"(1)] by blast
3656    AOT_hence ‹∀y ¬([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3657      using "cqt-further:4"[THEN "→E"] by blast
3658    AOT_hence ‹¬([λz z = a] = [λz z = a] & ¬a[λz z = a])›
3659      using "∀E" by blast
3660    AOT_hence ‹[λz z = a] = [λz z = a] → a[λz z = a]›
3661      by (metis "&I" "deduction-theorem" "raa-cor:4")
3662    AOT_hence ‹a[λz z = a]› using "=I"(1) θ[THEN "∀E"(2)] "→E" by blast
3663    AOT_thus ‹¬∀y [λz z = y]↓› using A RAA by blast
3664  qed
3665qed(simp)
3666
3667AOT_theorem "block-paradox:4": ‹¬∀y ∃F ∀x([F]x ≡ x = y)›
3668proof(rule RAA(2))
3669  AOT_assume θ: ‹∀y ∃F ∀x([F]x ≡ x = y)›
3670  AOT_have ‹∃x (A!x & ∀F (x[F] ≡ ∃z (∀y([F]y ≡ y = z) & ¬z[F])))›
3671    using "A-objects"[axiom_inst] by force
3672  then AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃z (∀y([F]y ≡ y = z) & ¬z[F]))›
3673    using "∃E"[rotated] by blast
3674  AOT_obtain F where F_prop: ‹∀x ([F]x ≡ x = a)› using θ[THEN "∀E"(2)] "∃E"[rotated] by blast
3675  AOT_have ζ: ‹a[F] ≡ ∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3676    using a_prop[THEN "&E"(2), THEN "∀E"(2)] by blast
3677  AOT_show ‹¬∀y ∃F ∀x([F]x ≡ x = y)›
3678  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3679    AOT_assume A: ‹a[F]›
3680    AOT_hence ‹∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3681      using ζ[THEN "≡E"(1)] by blast
3682    then AOT_obtain b where b_prop: ‹∀y ([F]y ≡ y = b) & ¬b[F]›
3683      using "∃E"[rotated] by blast
3684    moreover AOT_have ‹[F]a› using F_prop[THEN "∀E"(2), THEN "≡E"(2)] "=I"(2) by blast
3685    ultimately AOT_have ‹a = b› using "∀E"(2) "≡E"(1) "&E" by fast
3686    AOT_hence ‹a = b› using "β→C" by blast
3687    AOT_hence ‹b[F]› using A "rule=E" by fast
3688    AOT_thus ‹¬∀y ∃F ∀x([F]x ≡ x = y)› using b_prop[THEN "&E"(2)] RAA by blast
3689  next
3690    AOT_assume A: ‹¬a[F]›
3691    AOT_hence ‹¬∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3692      using ζ "oth-class-taut:4:b"[THEN "≡E"(1), THEN "≡E"(1)] by blast
3693    AOT_hence ‹∀z ¬(∀y ([F]y ≡ y = z) & ¬z[F])›
3694      using "cqt-further:4"[THEN "→E"] by blast
3695    AOT_hence ‹¬(∀y ([F]y ≡ y = a) & ¬a[F])›
3696      using "∀E" by blast
3697    AOT_hence ‹∀y ([F]y ≡ y = a) → a[F]›
3698      by (metis "&I" "deduction-theorem" "raa-cor:4")
3699    AOT_hence ‹a[F]› using F_prop "→E" by blast
3700    AOT_thus ‹¬∀y ∃F ∀x([F]x ≡ x = y)› using A RAA by blast
3701  qed
3702qed(simp)
3703
3704AOT_theorem "block-paradox:5": ‹¬∃F∀x∀y([F]xy ≡ y = x)›
3705proof(rule "raa-cor:2")
3706  AOT_assume ‹∃F∀x∀y([F]xy ≡ y = x)›
3707  then AOT_obtain F where F_prop: ‹∀x∀y([F]xy ≡ y = x)› using "∃E"[rotated] by blast
3708  {
3709    fix x
3710    AOT_have 1: ‹∀y([F]xy ≡ y = x)› using F_prop "∀E" by blast
3711    AOT_have 2: ‹[λz [F]xz]↓› by "cqt:2[lambda]"
3712    moreover AOT_have ‹∀y([λz [F]xz]y ≡ y = x)›
3713    proof(rule "∀I")
3714      fix y
3715      AOT_have ‹[λz [F]xz]y ≡ [F]xy›
3716        using "beta-C-meta"[THEN "→E"] 2 by fast
3717      also AOT_have ‹... ≡ y = x› using 1 "∀E"
3718        by fast
3719      finally AOT_show ‹[λz [F]xz]y ≡ y = x›.
3720    qed
3721    ultimately AOT_have ‹∃F∀y([F]y ≡ y = x)›
3722      using "∃I" by fast
3723  }
3724  AOT_hence ‹∀x∃F∀y([F]y ≡ y = x)›
3725    by (rule GEN)
3726  AOT_thus ‹∀x∃F∀y([F]y ≡ y = x) & ¬∀x∃F∀y([F]y ≡ y = x)›
3727    using "&I" "block-paradox:4" by blast
3728qed
3729
3730AOT_act_theorem "block-paradox2:1": ‹∀x [G]x → ¬[λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3731proof(rule "→I"; rule "raa-cor:2")
3732  AOT_assume antecedant: ‹∀x [G]x›
3733  AOT_have Lemma: ‹∀x ([G]ιy(y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃H (x[H] & ¬[H]x))›
3734  proof(rule GEN)
3735    fix x
3736    AOT_have A: ‹[G]ιy (y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3737    proof(rule "≡I"; rule "→I")
3738      AOT_assume ‹[G]ιy (y = x & ∃H (x[H] & ¬[H]x))›
3739      AOT_hence ‹ιy (y = x & ∃H (x[H] & ¬[H]x))↓›
3740        using "cqt:5:a"[axiom_inst, THEN "→E", THEN "&E"(2)] by blast
3741      AOT_thus ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3742        using "1-exists:1"[THEN "≡E"(1)] by blast
3743    next
3744      AOT_assume A: ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3745      AOT_obtain a where a_1: ‹a = x & ∃H (x[H] & ¬[H]x)› and a_2: ‹∀z (z = x & ∃H (x[H] & ¬[H]x) → z = a)›
3746        using "uniqueness:1"[THEN "≡dfE", OF A] "&E" "∃E"[rotated] by blast
3747      AOT_have a_3: ‹[G]a›
3748        using antecedant "∀E" by blast
3749      AOT_show ‹[G]ιy (y = x & ∃H (x[H] & ¬[H]x))›
3750        apply (rule "russell-axiom[exe,1].russell-axiom"[THEN "≡E"(2)])
3751        apply (rule "∃I"(2))
3752        using a_1 a_2 a_3 "&I" by blast
3753    qed
3754    also AOT_have B: ‹... ≡ ∃H (x[H] & ¬[H]x)›
3755    proof (rule "≡I"; rule "→I")
3756      AOT_assume A: ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3757      AOT_obtain a where ‹a = x & ∃H (x[H] & ¬[H]x)›
3758        using "uniqueness:1"[THEN "≡dfE", OF A] "&E" "∃E"[rotated] by blast
3759      AOT_thus ‹∃H (x[H] & ¬[H]x)› using "&E" by blast
3760    next
3761      AOT_assume ‹∃H (x[H] & ¬[H]x)›
3762      AOT_hence ‹x = x & ∃H (x[H] & ¬[H]x)›
3763        using "id-eq:1" "&I" by blast
3764      moreover AOT_have ‹∀z (z = x & ∃H (x[H] & ¬[H]x) → z = x)›
3765        by (simp add: "Conjunction Simplification"(1) "universal-cor")
3766      ultimately AOT_show ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3767        using "uniqueness:1"[THEN "≡dfI"] "&I" "∃I"(2) by fast
3768    qed
3769    finally AOT_show ‹([G]ιy(y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃H (x[H] & ¬[H]x))› .
3770  qed
3771
3772  AOT_assume A: ‹[λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3773  AOT_have θ: ‹∀x ([λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]x ≡ [G]ιy(y = x & ∃H (x[H] & ¬[H]x)))›
3774    using "beta-C-meta"[THEN "→E", OF A] "∀I" by fast
3775  AOT_have ‹∀x ([λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]x ≡ ∃H (x[H] & ¬[H]x))›
3776    using θ Lemma "cqt-basic:10"[THEN "→E"] "&I" by fast
3777  AOT_hence ‹∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x))›
3778    using "∃I"(1) A by fast
3779  AOT_thus ‹(∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x))) & (¬∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x)))›
3780    using "block-paradox:2" "&I" by blast
3781qed
3782
3783AOT_act_theorem "block-paradox2:2": ‹∃G ¬[λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3784proof(rule "∃I"(1))
3785  AOT_have 0: ‹[λx ∀p (p →p)]↓›
3786    by "cqt:2[lambda]"
3787  moreover AOT_have ‹∀x [λx ∀p (p →p)]x›
3788    apply (rule GEN)
3789    apply (rule "beta-C-cor:2"[THEN "→E", OF 0, THEN "∀E"(2), THEN "≡E"(2)])
3790    using "if-p-then-p" GEN by fast
3791  moreover AOT_have ‹∀G (∀x [G]x → ¬[λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓)›
3792      using "block-paradox2:1" "∀I" by fast
3793  ultimately AOT_show ‹¬[λx [λx ∀p (p →p)]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3794    using "∀E"(1) "→E" by blast
3795qed("cqt:2[lambda]")
3796
3797AOT_theorem propositions: ‹∃p □(p ≡ φ)›
3798proof(rule "∃I"(1))
3799  AOT_show ‹□(φ ≡ φ)›
3800    by (simp add: RN "oth-class-taut:3:a")
3801next
3802  AOT_show ‹φ↓›
3803    by (simp add: "log-prop-prop:2")
3804qed
3805
3806AOT_theorem "pos-not-equiv-ne:1": ‹(◇¬∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)) → F ≠ G›
3807proof (rule "→I")
3808  AOT_assume ‹◇¬∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)›
3809  AOT_hence ‹¬□∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)›
3810    using "KBasic:11"[THEN "≡E"(2)] by blast
3811  AOT_hence ‹¬(F = G)›
3812    using "id-rel-nec-equiv:1" "modus-tollens:1" by blast
3813  AOT_thus ‹F ≠ G›
3814    using "=-infix"[THEN "≡dfI"] by blast
3815qed
3816
3817AOT_theorem "pos-not-equiv-ne:2": ‹(◇¬(φ{F} ≡ φ{G})) → F ≠ G›
3818proof (rule "→I")
3819  AOT_modally_strict {
3820    AOT_have ‹¬(φ{F} ≡ φ{G}) → ¬(F = G)›
3821    proof (rule "→I"; rule "raa-cor:2")
3822      AOT_assume 1: ‹F = G›
3823      AOT_hence ‹φ{F} → φ{G}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3824      moreover {
3825        AOT_have ‹G = F› using 1 id_sym by blast
3826        AOT_hence ‹φ{G} → φ{F}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3827      }
3828      ultimately AOT_have ‹φ{F} ≡ φ{G}› using "≡I" by blast
3829      moreover AOT_assume ‹¬(φ{F} ≡ φ{G})›
3830      ultimately AOT_show ‹(φ{F} ≡ φ{G}) & ¬(φ{F} ≡ φ{G})›
3831        using "&I" by blast
3832    qed
3833  }
3834  AOT_hence ‹◇¬(φ{F} ≡ φ{G}) → ◇¬(F = G)›
3835    using "RM:2[prem]" by blast
3836  moreover AOT_assume ‹◇¬(φ{F} ≡ φ{G})›
3837  ultimately AOT_have 0: ‹◇¬(F = G)› using "→E" by blast
3838  AOT_have ‹◇(F ≠ G)›
3839    by (AOT_subst "«F ≠ G»" "«¬(F = G)»")
3840       (auto simp: "=-infix" "≡Df" 0)
3841  AOT_thus ‹F ≠ G›
3842    using "id-nec2:3"[THEN "→E"] by blast
3843qed
3844
3845AOT_theorem "pos-not-equiv-ne:2[zero]": ‹(◇¬(φ{p} ≡ φ{q})) → p ≠ q›
3846proof (rule "→I")
3847  AOT_modally_strict {
3848    AOT_have ‹¬(φ{p} ≡ φ{q}) → ¬(p = q)›
3849    proof (rule "→I"; rule "raa-cor:2")
3850      AOT_assume 1: ‹p = q›
3851      AOT_hence ‹φ{p} → φ{q}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3852      moreover {
3853        AOT_have ‹q = p› using 1 id_sym by blast
3854        AOT_hence ‹φ{q} → φ{p}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3855      }
3856      ultimately AOT_have ‹φ{p} ≡ φ{q}› using "≡I" by blast
3857      moreover AOT_assume ‹¬(φ{p} ≡ φ{q})›
3858      ultimately AOT_show ‹(φ{p} ≡ φ{q}) & ¬(φ{p} ≡ φ{q})›
3859        using "&I" by blast
3860    qed
3861  }
3862  AOT_hence ‹◇¬(φ{p} ≡ φ{q}) → ◇¬(p = q)›
3863    using "RM:2[prem]" by blast
3864  moreover AOT_assume ‹◇¬(φ{p} ≡ φ{q})›
3865  ultimately AOT_have 0: ‹◇¬(p = q)› using "→E" by blast
3866  AOT_have ‹◇(p ≠ q)›
3867    by (AOT_subst "«p ≠ q»" "«¬(p = q)»")
3868       (auto simp: 0 "=-infix" "≡Df")
3869  AOT_thus ‹p ≠ q›
3870    using "id-nec2:3"[THEN "→E"] by blast
3871qed
3872
3873AOT_theorem "pos-not-equiv-ne:3": ‹(¬∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)) → F ≠ G›
3874  using "→I" "pos-not-equiv-ne:1"[THEN "→E"] "T◇"[THEN "→E"] by blast
3875
3876AOT_theorem "pos-not-equiv-ne:4": ‹(¬(φ{F} ≡ φ{G})) → F ≠ G›
3877  using "→I" "pos-not-equiv-ne:2"[THEN "→E"] "T◇"[THEN "→E"] by blast
3878
3879AOT_theorem "pos-not-equiv-ne:4[zero]": ‹(¬(φ{p} ≡ φ{q})) → p ≠ q›
3880  using "→I" "pos-not-equiv-ne:2[zero]"[THEN "→E"] "T◇"[THEN "→E"] by blast
3881
3882AOT_define relation_negation :: "Π ⇒ Π" ("_-")
3883  "df-relation-negation": "[F]- =df [λx1...xn ¬[F]x1...xn]"
3884
3885nonterminal φneg
3886syntax "" :: "φneg ⇒ τ" ("_")
3887syntax "" :: "φneg ⇒ φ" ("'(_')")
3888
3889AOT_define relation_negation_0 :: ‹φ ⇒ φneg› ("'(_')-")
3890  "df-relation-negation[zero]": "(p)- =df [λ ¬p]"
3891
3892AOT_theorem "rel-neg-T:1": ‹[λx1...xn ¬[Π]x1...xn]↓›
3893  by "cqt:2[lambda]"
3894
3895AOT_theorem "rel-neg-T:1[zero]": ‹[λ ¬φ]↓›
3896  using "cqt:2[lambda0]"[axiom_inst] by blast
3897
3898AOT_theorem "rel-neg-T:2": ‹[Π]- = [λx1...xn ¬[Π]x1...xn]›
3899  using "=I"(1)[OF "rel-neg-T:1"]
3900  by (rule "=dfI"(1)[OF "df-relation-negation", OF "rel-neg-T:1"])
3901
3902AOT_theorem "rel-neg-T:2[zero]": ‹(φ)- = [λ ¬φ]›
3903  using "=I"(1)[OF "rel-neg-T:1[zero]"]
3904  by (rule "=dfI"(1)[OF "df-relation-negation[zero]", OF "rel-neg-T:1[zero]"])
3905
3906AOT_theorem "rel-neg-T:3": ‹[Π]-↓›
3907  using "=dfI"(1)[OF "df-relation-negation", OF "rel-neg-T:1"] "rel-neg-T:1" by blast
3908
3909AOT_theorem "rel-neg-T:3[zero]": ‹(φ)-↓›
3910  using "log-prop-prop:2" by blast
3911(*  using "=dfI"(1)[OF "df-relation-negation[zero]", OF "rel-neg-T:1[zero]"] "rel-neg-T:1[zero]" by blast *)
3912
3913(* Note: PLM states the zero place case twice *)
3914AOT_theorem "thm-relation-negation:1": ‹[F]-x1...xn ≡ ¬[F]x1...xn›
3915proof -
3916  AOT_have ‹[F]-x1...xn ≡ [λx1...xn ¬[F]x1...xn]x1...xn›
3917    using "rule=E"[rotated, OF "rel-neg-T:2"] "rule=E"[rotated, OF "rel-neg-T:2"[THEN id_sym]]
3918    "→I" "≡I" by fast
3919  also AOT_have ‹... ≡ ¬[F]x1...xn›
3920    using "beta-C-meta"[THEN "→E", OF "rel-neg-T:1"] by fast
3921  finally show ?thesis.
3922qed
3923
3924AOT_theorem "thm-relation-negation:2": ‹¬[F]-x1...xn ≡ [F]x1...xn›
3925  apply (AOT_subst "«[F]x1...xn»" "«¬¬[F]x1...xn»")
3926   apply (simp add: "oth-class-taut:3:b")
3927  apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
3928  using "thm-relation-negation:1".
3929
3930AOT_theorem "thm-relation-negation:3": ‹((p)-) ≡ ¬p›
3931proof -
3932  AOT_have ‹(p)- = [λ ¬p]› using "rel-neg-T:2[zero]" by blast
3933  AOT_hence ‹((p)-) ≡ [λ ¬p]›
3934    using "df-relation-negation[zero]" "log-prop-prop:2" "oth-class-taut:3:a" "rule-id-def:2:a" by blast
3935  also AOT_have ‹[λ ¬p] ≡ ¬p›
3936    by (simp add: "propositions-lemma:2")
3937  finally show ?thesis.
3938qed
3939
3940AOT_theorem "thm-relation-negation:4": ‹(¬((p)-)) ≡ p›
3941  using "thm-relation-negation:3"[THEN "≡E"(1)]
3942        "thm-relation-negation:3"[THEN "≡E"(2)]
3943        "≡I" "→I" RAA by metis
3944
3945AOT_theorem "thm-relation-negation:5": ‹[F] ≠ [F]-›
3946proof -
3947  AOT_have ‹¬([F] = [F]-)›
3948  proof (rule RAA(2))
3949    AOT_show ‹[F]x1...xn → [F]x1...xn› for x1xn
3950      using "if-p-then-p".
3951  next
3952    AOT_assume ‹[F] = [F]-›
3953    AOT_hence ‹[F]- = [F]› using id_sym by blast
3954    AOT_hence ‹[F]x1...xn ≡ ¬[F]x1...xn› for x1xn
3955      using "rule=E" "thm-relation-negation:1" by fast
3956    AOT_thus ‹¬([F]x1...xn → [F]x1...xn)› for x1xn
3957      using "≡E" RAA by metis
3958  qed
3959  thus ?thesis
3960    using "≡dfI" "=-infix" by blast
3961qed
3962
3963AOT_theorem "thm-relation-negation:6": ‹p ≠ (p)-›
3964proof -
3965  AOT_have ‹¬(p = (p)-)›
3966  proof (rule RAA(2))
3967    AOT_show ‹p → p›
3968      using "if-p-then-p".
3969  next
3970    AOT_assume ‹p = (p)-›
3971    AOT_hence ‹(p)- = p› using id_sym by blast
3972    AOT_hence ‹p ≡ ¬p›
3973      using "rule=E" "thm-relation-negation:3" by fast
3974    AOT_thus ‹¬(p → p)›
3975      using "≡E" RAA by metis
3976  qed
3977  thus ?thesis
3978    using "≡dfI" "=-infix" by blast
3979qed
3980
3981AOT_theorem "thm-relation-negation:7": ‹(p)- = (¬p)›
3982  apply (rule "df-relation-negation[zero]"[THEN "=dfE"(1)])
3983  using "cqt:2[lambda0]"[axiom_inst] "rel-neg-T:2[zero]" "propositions-lemma:1" id_trans by blast+
3984
3985AOT_theorem "thm-relation-negation:8": ‹p = q → (¬p) = (¬q)›
3986proof(rule "→I")
3987  AOT_assume ‹p = q›
3988  moreover AOT_have ‹(¬p)↓› using "log-prop-prop:2".
3989  moreover AOT_have ‹(¬p) = (¬p)› using calculation(2) "=I" by blast
3990  ultimately AOT_show ‹(¬p) = (¬q)›
3991    using "rule=E" by fast
3992qed
3993
3994AOT_theorem "thm-relation-negation:9": ‹p = q → (p)- = (q)-›
3995proof(rule "→I")
3996  AOT_assume ‹p = q›
3997  AOT_hence ‹(¬p) = (¬q)› using "thm-relation-negation:8" "→E" by blast
3998  AOT_thus ‹(p)- = (q)-›
3999    using "thm-relation-negation:7" id_sym id_trans by metis
4000qed
4001
4002AOT_define Necessary :: ‹Π ⇒ φ› ("Necessary'(_')")
4003  "contingent-properties:1": ‹Necessary([F]) ≡df □∀x1...∀xn [F]x1...xn›
4004
4005AOT_define Necessary0 :: ‹φ ⇒ φ› ("Necessary0'(_')")
4006  "contingent-properties:1[zero]": ‹Necessary0(p) ≡df □p›
4007
4008AOT_define Impossible :: ‹Π ⇒ φ› ("Impossible'(_')")
4009  "contingent-properties:2": ‹Impossible([F]) ≡df F↓ & □∀x1...∀xn ¬[F]x1...xn›
4010
4011AOT_define Impossible0 :: ‹φ ⇒ φ› ("Impossible0'(_')")
4012  "contingent-properties:2[zero]": ‹Impossible0(p) ≡df □¬p›
4013
4014AOT_define NonContingent :: ‹Π ⇒ φ› ("NonContingent'(_')")
4015  "contingent-properties:3": ‹NonContingent([F]) ≡df Necessary([F]) ∨ Impossible([F])›
4016
4017AOT_define NonContingent0 :: ‹φ ⇒ φ› ("NonContingent0'(_')")
4018  "contingent-properties:3[zero]": ‹NonContingent0(p) ≡df Necessary0(p) ∨ Impossible0(p)›
4019
4020AOT_define Contingent :: ‹Π ⇒ φ› ("Contingent'(_')")
4021  "contingent-properties:4": ‹Contingent([F]) ≡df F↓ & ¬(Necessary([F]) ∨ Impossible([F]))›
4022
4023AOT_define Contingent0 :: ‹φ ⇒ φ› ("Contingent0'(_')")
4024  "contingent-properties:4[zero]": ‹Contingent0(p) ≡df ¬(Necessary0(p) ∨ Impossible0(p))›
4025
4026
4027AOT_theorem "thm-cont-prop:1": ‹NonContingent([F]) ≡ NonContingent([F]-)›
4028proof (rule "≡I"; rule "→I")
4029  AOT_assume ‹NonContingent([F])›
4030  AOT_hence ‹Necessary([F]) ∨ Impossible([F])›
4031    using "≡dfE"[OF "contingent-properties:3"] by blast
4032  moreover {
4033    AOT_assume ‹Necessary([F])›
4034    AOT_hence ‹□(∀x1...∀xn [F]x1...xn)›
4035      using "≡dfE"[OF "contingent-properties:1"] by blast
4036    moreover AOT_modally_strict {
4037      AOT_assume ‹∀x1...∀xn [F]x1...xn›
4038      AOT_hence ‹[F]x1...xn› for x1xn using "∀E" by blast
4039      AOT_hence ‹¬[F]-x1...xn› for x1xn
4040        by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:2" "≡E"(1))
4041      AOT_hence ‹∀x1...∀xn ¬[F]-x1...xn› using "∀I" by fast
4042    }
4043    ultimately AOT_have ‹□(∀x1...∀xn ¬[F]-x1...xn)›
4044      using "RN[prem]"[where Γ="{«∀x1...∀xn [F]x1...xn»}", simplified] by blast
4045    AOT_hence ‹Impossible([F]-)›
4046      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "rel-neg-T:3", THEN "≡E"(2)]
4047      by blast
4048  }
4049  moreover {
4050    AOT_assume ‹Impossible([F])›
4051    AOT_hence ‹□(∀x1...∀xn ¬[F]x1...xn)›
4052      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)]
4053      by blast
4054    moreover AOT_modally_strict {
4055      AOT_assume ‹∀x1...∀xn ¬[F]x1...xn›
4056      AOT_hence ‹¬[F]x1...xn› for x1xn using "∀E" by blast
4057      AOT_hence ‹[F]-x1...xn› for x1xn
4058        by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:1" "≡E"(1))
4059      AOT_hence ‹∀x1...∀xn [F]-x1...xn› using "∀I" by fast
4060    }
4061    ultimately AOT_have ‹□(∀x1...∀xn [F]-x1...xn)›
4062      using "RN[prem]"[where Γ="{«∀x1...∀xn ¬[F]x1...xn»}"] by blast
4063    AOT_hence ‹Necessary([F]-)›
4064      using "≡dfI"[OF "contingent-properties:1"] by blast
4065  }
4066  ultimately AOT_have ‹Necessary([F]-) ∨ Impossible([F]-)›
4067    using "∨E"(1) "∨I" "→I" by metis
4068  AOT_thus ‹NonContingent([F]-)›
4069    using "≡dfI"[OF "contingent-properties:3"] by blast
4070next
4071  AOT_assume ‹NonContingent([F]-)›
4072  AOT_hence ‹Necessary([F]-) ∨ Impossible([F]-)›
4073    using "≡dfE"[OF "contingent-properties:3"] by blast
4074  moreover {
4075    AOT_assume ‹Necessary([F]-)›
4076    AOT_hence ‹□(∀x1...∀xn [F]-x1...xn)›
4077      using "≡dfE"[OF "contingent-properties:1"] by blast
4078    moreover AOT_modally_strict {
4079      AOT_assume ‹∀x1...∀xn [F]-x1...xn›
4080      AOT_hence ‹[F]-x1...xn› for x1xn using "∀E" by blast
4081      AOT_hence ‹¬[F]x1...xn› for x1xn
4082        by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:1" "≡E"(2))
4083      AOT_hence ‹∀x1...∀xn ¬[F]x1...xn› using "∀I" by fast
4084    }
4085    ultimately AOT_have ‹□∀x1...∀xn ¬[F]x1...xn›
4086      using "RN[prem]"[where Γ="{«∀x1...∀xn [F]-x1...xn»}"] by blast
4087    AOT_hence ‹Impossible([F])›
4088      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(2)]
4089      by blast
4090  }
4091  moreover {
4092    AOT_assume ‹Impossible([F]-)›
4093    AOT_hence ‹□(∀x1...∀xn ¬[F]-x1...xn)›
4094      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "rel-neg-T:3", THEN "≡E"(1)]
4095      by blast
4096    moreover AOT_modally_strict {
4097      AOT_assume ‹∀x1...∀xn ¬[F]-x1...xn›
4098      AOT_hence ‹¬[F]-x1...xn› for x1xn using "∀E" by blast
4099      AOT_hence ‹[F]x1...xn› for x1xn 
4100        using "thm-relation-negation:1"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4101              "useful-tautologies:1"[THEN "→E"] by blast
4102      AOT_hence ‹∀x1...∀xn [F]x1...xn› using "∀I" by fast
4103    }
4104    ultimately AOT_have ‹□(∀x1...∀xn [F]x1...xn)›
4105      using "RN[prem]"[where Γ="{«∀x1...∀xn ¬[F]-x1...xn»}"] by blast
4106    AOT_hence ‹Necessary([F])›
4107      using "≡dfI"[OF "contingent-properties:1"] by blast
4108  }
4109  ultimately AOT_have ‹Necessary([F]) ∨ Impossible([F])›
4110    using "∨E"(1) "∨I" "→I" by metis
4111  AOT_thus ‹NonContingent([F])›
4112    using "≡dfI"[OF "contingent-properties:3"] by blast
4113qed
4114
4115AOT_theorem "thm-cont-prop:2": ‹Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4116proof -
4117  AOT_have ‹Contingent([F]) ≡ ¬(Necessary([F]) ∨ Impossible([F]))›
4118    using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst]]
4119    by blast
4120  also AOT_have ‹... ≡ ¬Necessary([F]) & ¬Impossible([F])›
4121    using "oth-class-taut:5:d" by fastforce
4122  also AOT_have ‹... ≡ ¬Impossible([F]) & ¬Necessary([F])›
4123    by (simp add: "Commutativity of &")
4124  also AOT_have ‹... ≡ ◇∃x [F]x & ¬Necessary([F])›
4125  proof (rule "oth-class-taut:4:e"[THEN "→E"])
4126    AOT_have ‹¬Impossible([F]) ≡ ¬□¬ ∃x [F]x›
4127      apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4128      apply (AOT_subst "«∃x [F]x»" "«¬ ∀x ¬[F]x»")
4129       apply (simp add: "conventions:4" "≡Df")
4130      apply (AOT_subst_rev "«∀x ¬[F]x»" "«¬¬∀x ¬[F]x»" )
4131       apply (simp add: "oth-class-taut:3:b")
4132      using "contingent-properties:2"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst]] by blast
4133    also AOT_have ‹... ≡ ◇∃x [F]x›
4134      using "conventions:5"[THEN "≡Df", symmetric] by blast
4135    finally AOT_show ‹¬Impossible([F]) ≡ ◇∃x [F]x› .
4136  qed
4137  also AOT_have ‹... ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4138  proof (rule "oth-class-taut:4:f"[THEN "→E"])
4139    AOT_have ‹¬Necessary([F]) ≡ ¬□¬∃x ¬[F]x›
4140      apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4141      apply (AOT_subst "«∃x ¬[F]x»" "«¬ ∀x ¬¬[F]x»")
4142       apply (simp add: "conventions:4" "≡Df")
4143      apply (AOT_subst_rev "λ κ . «[F]κ»" "λ κ . «¬¬[F]κ»")
4144       apply (simp add: "oth-class-taut:3:b")
4145      apply (AOT_subst_rev "«∀x [F]x»" "«¬¬∀x [F]x»")
4146      by (auto simp: "oth-class-taut:3:b" "contingent-properties:1" "≡Df")
4147    also AOT_have ‹... ≡ ◇∃x ¬[F]x›
4148      using "conventions:5"[THEN "≡Df", symmetric] by blast
4149    finally AOT_show ‹¬Necessary([F]) ≡ ◇∃x ¬[F]x›.
4150  qed
4151  finally show ?thesis.
4152qed
4153
4154AOT_theorem "thm-cont-prop:3": ‹Contingent([F]) ≡ Contingent([F]-)› for F::‹<κ> AOT_var›
4155proof -
4156  {
4157    fix Π :: ‹<κ>›
4158    AOT_assume ‹Π↓›
4159    moreover AOT_have ‹∀F (Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x)›
4160      using "thm-cont-prop:2" GEN by fast
4161    ultimately  AOT_have ‹Contingent([Π]) ≡ ◇∃x [Π]x & ◇∃x ¬[Π]x›
4162      using "thm-cont-prop:2" "∀E" by fast
4163  } note 1 = this
4164  AOT_have ‹Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4165    using "thm-cont-prop:2" by blast
4166  also AOT_have ‹... ≡ ◇∃x ¬[F]x & ◇∃x [F]x›
4167    by (simp add: "Commutativity of &")
4168  also AOT_have ‹... ≡ ◇∃x [F]-x & ◇∃x [F]x›
4169    by (AOT_subst "λ κ . «[F]-κ»"  "λκ . «¬[F]κ»")
4170       (auto simp: "thm-relation-negation:1" "oth-class-taut:3:a")
4171  also AOT_have ‹... ≡ ◇∃x [F]-x & ◇∃x ¬[F]-x›
4172    by (AOT_subst_rev "λ κ . «¬[F]-κ»"  "λκ . «[F]κ»")
4173       (auto simp: "thm-relation-negation:2" "oth-class-taut:3:a")
4174  also AOT_have ‹... ≡ Contingent([F]-)›
4175    using 1[OF "rel-neg-T:3", symmetric] by blast
4176  finally show ?thesis.
4177qed
4178
4179AOT_define concrete_if_concrete :: ‹Π› ("L")  L_def: ‹L =df [λx E!x → E!x]›
4180
4181AOT_theorem "thm-noncont-e-e:1": ‹Necessary(L)›
4182proof -
4183  AOT_modally_strict {
4184    fix x
4185    AOT_have ‹[λx E!x → E!x]↓› by "cqt:2[lambda]"
4186    moreover AOT_have ‹x↓› using "cqt:2[const_var]"[axiom_inst] by blast
4187    moreover AOT_have ‹E!x → E!x› using "if-p-then-p" by blast
4188    ultimately AOT_have ‹[λx E!x → E!x]x›
4189      using "β←C" by blast
4190  }
4191  AOT_hence 0: ‹□∀x [λx E!x → E!x]x›
4192    using RN GEN by blast
4193  show ?thesis
4194    apply (rule "=dfI"(2)[OF L_def])
4195     apply "cqt:2[lambda]"
4196    by (rule "contingent-properties:1"[THEN "≡dfI", OF 0])
4197qed
4198
4199AOT_theorem "thm-noncont-e-e:2": ‹Impossible([L]-)›
4200proof -
4201  AOT_modally_strict {
4202    fix x
4203
4204    AOT_have 0: ‹∀F (¬[F]-x ≡ [F]x)›
4205      using "thm-relation-negation:2" GEN by fast
4206    AOT_have ‹¬[λx E!x → E!x]-x ≡ [λx E!x → E!x]x›
4207      by (rule 0[THEN "∀E"(1)]) "cqt:2[lambda]"
4208    moreover {
4209      AOT_have ‹[λx E!x → E!x]↓› by "cqt:2[lambda]"
4210      moreover AOT_have ‹x↓› using "cqt:2[const_var]"[axiom_inst] by blast
4211      moreover AOT_have ‹E!x → E!x› using "if-p-then-p" by blast
4212      ultimately AOT_have ‹[λx E!x → E!x]x›
4213        using "β←C" by blast
4214    }
4215    ultimately AOT_have ‹¬[λx E!x → E!x]-x›
4216      using "≡E" by blast
4217  }
4218  AOT_hence 0: ‹□∀x ¬[λx E!x → E!x]-x›
4219    using RN GEN by fast
4220  show ?thesis
4221    apply (rule "=dfI"(2)[OF L_def])
4222     apply "cqt:2[lambda]"
4223    apply (rule "contingent-properties:2"[THEN "≡dfI"]; rule "&I")
4224     using "rel-neg-T:3"
4225     apply blast
4226    using 0
4227    by blast
4228qed
4229
4230AOT_theorem "thm-noncont-e-e:3": ‹NonContingent(L)›
4231  using "thm-noncont-e-e:1"
4232  by (rule "contingent-properties:3"[THEN "≡dfI", OF "∨I"(1)])
4233
4234AOT_theorem "thm-noncont-e-e:4": ‹NonContingent([L]-)›
4235proof -
4236  AOT_have 0: ‹∀F (NonContingent([F]) ≡ NonContingent([F]-))›
4237    using "thm-cont-prop:1" "∀I" by fast
4238  moreover AOT_have 1: ‹L↓›
4239    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
4240  AOT_show ‹NonContingent([L]-)›
4241    using "∀E"(1)[OF 0, OF 1, THEN "≡E"(1), OF "thm-noncont-e-e:3"] by blast
4242qed
4243
4244AOT_theorem "thm-noncont-e-e:5": ‹∃F ∃G (F ≠ «G::<κ>» & NonContingent([F]) & NonContingent([G]))›
4245proof (rule "∃I")+
4246  {
4247    AOT_have ‹∀F [F] ≠ [F]-› using "thm-relation-negation:5" GEN by fast
4248    moreover AOT_have ‹L↓›
4249      by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
4250    ultimately AOT_have ‹L ≠ [L]-› using "∀E" by blast
4251  }
4252  AOT_thus ‹L ≠ [L]- & NonContingent(L) & NonContingent([L]-)›
4253    using "thm-noncont-e-e:3" "thm-noncont-e-e:4" "&I" by metis
4254next
4255  AOT_show ‹[L]-↓›
4256    using "rel-neg-T:3" by blast
4257next
4258  AOT_show ‹L↓›
4259      by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
4260qed
4261
4262AOT_theorem "lem-cont-e:1": ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x)›
4263proof -
4264  AOT_have ‹◇∃x ([F]x & ◇¬[F]x) ≡ ∃x ◇([F]x & ◇¬[F]x)›
4265    using "BF◇" "CBF◇" "≡I" by blast
4266  also AOT_have ‹… ≡ ∃x (◇[F]x &  ◇¬[F]x)›
4267    by (AOT_subst ‹λκ. «◇([F]κ & ◇¬[F]κ)»›  ‹λ κ .  «◇[F]κ &  ◇¬[F]κ»›)
4268       (auto simp: "S5Basic:11" "cqt-further:7")
4269  also AOT_have ‹… ≡ ∃x (◇¬[F]x & ◇[F]x)›
4270    by (AOT_subst ‹λκ. «◇¬[F]κ & ◇[F]κ»›  ‹λ κ .  «◇[F]κ & ◇¬[F]κ»›)
4271       (auto simp: "Commutativity of &" "cqt-further:7")
4272  also AOT_have ‹… ≡ ∃x ◇(¬[F]x & ◇[F]x)›
4273    by (AOT_subst ‹λ κ .  «◇(¬[F]κ & ◇[F]κ)»› ‹λκ. «◇¬[F]κ & ◇[F]κ»›)
4274       (auto simp: "S5Basic:11" "oth-class-taut:3:a")
4275  also AOT_have ‹… ≡ ◇∃x (¬[F]x & ◇[F]x)›
4276    using "BF◇" "CBF◇" "≡I" by fast
4277  finally show ?thesis.
4278qed
4279
4280AOT_theorem "lem-cont-e:2": ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x ([F]-x & ◇¬[F]-x)›
4281proof -
4282  AOT_have ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x)›
4283    using "lem-cont-e:1".
4284  also AOT_have ‹… ≡ ◇∃x ([F]-x & ◇¬[F]-x)›
4285    apply (AOT_subst "λ κ . «¬[F]-κ»" "λ κ . «[F]κ»")
4286     apply (simp add: "thm-relation-negation:2")
4287    apply (AOT_subst "λ κ . «[F]-κ»" "λ κ . «¬[F]κ»")
4288     apply (simp add: "thm-relation-negation:1")
4289    by (simp add: "oth-class-taut:3:a")
4290  finally show ?thesis.
4291qed
4292
4293AOT_theorem "thm-cont-e:1": ‹◇∃x (E!x & ◇¬E!x)›
4294proof (rule "CBF◇"[THEN "→E"])
4295  AOT_have ‹∃x ◇(E!x & ¬𝒜E!x)› using "qml:4"[axiom_inst] "BF◇"[THEN "→E"] by blast
4296  then AOT_obtain a where ‹◇(E!a & ¬𝒜E!a)› using "∃E"[rotated] by blast
4297  AOT_hence θ: ‹◇E!a & ◇¬𝒜E!a›
4298    using "KBasic2:3"[THEN "→E"] by blast
4299  AOT_have ξ: ‹◇E!a & ◇𝒜¬E!a›
4300    by (AOT_subst  "«𝒜¬E!a»" "«¬𝒜E!a»")
4301       (auto simp: "logic-actual-nec:1"[axiom_inst] θ)
4302  AOT_have ζ: ‹◇E!a & 𝒜¬E!a›
4303    by (AOT_subst "«𝒜¬E!a»" "«◇𝒜¬E!a»")
4304       (auto simp add: "Act-Sub:4" ξ)
4305  AOT_hence ‹◇E!a & ◇¬E!a›
4306    using "&E" "&I" "Act-Sub:3"[THEN "→E"] by blast
4307  AOT_hence ‹◇(E!a & ◇¬E!a)› using "S5Basic:11"[THEN "≡E"(2)] by simp
4308  AOT_thus ‹∃x ◇(E!x & ◇¬E!x)› using "∃I"(2) by fast
4309qed
4310
4311AOT_theorem "thm-cont-e:2": ‹◇∃x (¬E!x & ◇E!x)›
4312proof -
4313  AOT_have ‹∀F (◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x))›
4314    using "lem-cont-e:1" GEN by fast
4315  AOT_hence ‹(◇∃x (E!x & ◇¬E!x) ≡ ◇∃x (¬E!x & ◇E!x))›
4316    using "∀E"(1) "cqt:2[concrete]"[axiom_inst] by blast
4317  thus ?thesis using "thm-cont-e:1" "≡E" by blast
4318qed
4319
4320AOT_theorem "thm-cont-e:3": ‹◇∃x E!x›
4321proof (rule "CBF◇"[THEN "→E"])
4322  AOT_obtain a where ‹◇(E!a & ◇¬E!a)›
4323    using "∃E"[rotated, OF "thm-cont-e:1"[THEN "BF◇"[THEN "→E"]]] by blast
4324  AOT_hence ‹◇E!a›
4325    using "KBasic2:3"[THEN "→E", THEN "&E"(1)] by blast
4326  AOT_thus ‹∃x ◇E!x› using "∃I" by fast
4327qed
4328
4329AOT_theorem "thm-cont-e:4": ‹◇∃x ¬E!x›
4330proof (rule "CBF◇"[THEN "→E"])
4331  AOT_obtain a where ‹◇(E!a & ◇¬E!a)›
4332    using "∃E"[rotated, OF "thm-cont-e:1"[THEN "BF◇"[THEN "→E"]]] by blast
4333  AOT_hence ‹◇◇¬E!a›
4334    using "KBasic2:3"[THEN "→E", THEN "&E"(2)] by blast
4335  AOT_hence ‹◇¬E!a›
4336    using "4◇"[THEN "→E"] by blast
4337  AOT_thus ‹∃x ◇¬E!x› using "∃I" by fast
4338qed
4339
4340AOT_theorem "thm-cont-e:5": ‹Contingent([E!])›
4341proof -
4342  AOT_have ‹∀F (Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x)›
4343    using "thm-cont-prop:2" GEN by fast
4344  AOT_hence ‹Contingent([E!]) ≡ ◇∃x E!x & ◇∃x ¬E!x›
4345    using "∀E"(1) "cqt:2[concrete]"[axiom_inst] by blast
4346  thus ?thesis
4347    using "thm-cont-e:3" "thm-cont-e:4" "≡E"(2) "&I" by blast
4348qed
4349
4350AOT_theorem "thm-cont-e:6": ‹Contingent([E!]-)›
4351proof -
4352  AOT_have ‹∀F (Contingent([«F::<κ>»]) ≡ Contingent([F]-))›
4353    using "thm-cont-prop:3" GEN by fast
4354  AOT_hence ‹Contingent([E!]) ≡ Contingent([E!]-)›
4355    using "∀E" "cqt:2[concrete]"[axiom_inst] by fast
4356  thus ?thesis using "thm-cont-e:5" "≡E" by blast
4357qed
4358
4359AOT_theorem "thm-cont-e:7": ‹∃F∃G (Contingent([«F::<κ>»]) & Contingent([G]) & F ≠ G)›
4360proof (rule "∃I")+
4361  AOT_have ‹∀F [«F::<κ>»] ≠ [F]-› using "thm-relation-negation:5" GEN by fast
4362  AOT_hence ‹[E!] ≠ [E!]-›
4363    using "∀E" "cqt:2[concrete]"[axiom_inst] by fast
4364  AOT_thus ‹Contingent([E!]) & Contingent([E!]-) & [E!] ≠ [E!]-›
4365    using "thm-cont-e:5" "thm-cont-e:6" "&I" by metis
4366next
4367  AOT_show ‹E!-↓›
4368    by (fact AOT)
4369next
4370  AOT_show ‹E!↓› by (fact "cqt:2[concrete]"[axiom_inst])
4371qed
4372
4373AOT_theorem "property-facts:1": ‹NonContingent([F]) → ¬∃G (Contingent([G]) & G = F)›
4374proof (rule "→I"; rule "raa-cor:2")
4375  AOT_assume ‹NonContingent([F])›
4376  AOT_hence 1: ‹Necessary([F]) ∨ Impossible([F])›
4377    using "contingent-properties:3"[THEN "≡dfE"] by blast
4378  AOT_assume ‹∃G (Contingent([G]) & G = F)›
4379  then AOT_obtain G where ‹Contingent([G]) & G = F› using "∃E"[rotated] by blast
4380  AOT_hence ‹Contingent([F])› using "rule=E" "&E" by blast
4381  AOT_hence ‹¬(Necessary([F]) ∨ Impossible([F]))›
4382    using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)] by blast
4383  AOT_thus ‹(Necessary([F]) ∨ Impossible([F])) & ¬(Necessary([F]) ∨ Impossible([F]))›
4384    using 1 "&I" by blast
4385qed
4386
4387AOT_theorem "property-facts:2": ‹Contingent([F]) → ¬∃G (NonContingent([G]) & G = F)›
4388proof (rule "→I"; rule "raa-cor:2")
4389  AOT_assume ‹Contingent([F])›
4390  AOT_hence 1: ‹¬(Necessary([F]) ∨ Impossible([F]))›
4391    using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)] by blast
4392  AOT_assume ‹∃G (NonContingent([G]) & G = F)›
4393  then AOT_obtain G where ‹NonContingent([G]) & G = F› using "∃E"[rotated] by blast
4394  AOT_hence ‹NonContingent([F])› using "rule=E" "&E" by blast
4395  AOT_hence ‹Necessary([F]) ∨ Impossible([F])›
4396    using "contingent-properties:3"[THEN "≡dfE"] by blast
4397  AOT_thus ‹(Necessary([F]) ∨ Impossible([F])) & ¬(Necessary([F]) ∨ Impossible([F]))›
4398    using 1 "&I" by blast
4399qed
4400
4401AOT_theorem "property-facts:3": ‹L ≠ [L]- & L ≠ E! & L ≠ E!- & [L]- ≠ [E!]- & E! ≠ [E!]-›
4402proof -
4403  AOT_have noneqI: ‹Π ≠ Π'› if ‹φ{Π}› and ‹¬φ{Π'}› for φ Π Π'
4404    apply (rule "=-infix"[THEN "≡dfI"]; rule "raa-cor:2")
4405    using "rule=E"[where φ=φ and τ=Π and σ = Π'] that "&I" by blast
4406  AOT_have contingent_denotes: ‹Π↓› if ‹Contingent([Π])› for Π
4407    using that "contingent-properties:4"[THEN "≡dfE", THEN "&E"(1)] by blast
4408  AOT_have not_noncontingent_if_contingent: ‹¬NonContingent([Π])› if ‹Contingent([Π])› for Π
4409  proof(rule RAA(2))
4410    AOT_show ‹¬(Necessary([Π]) ∨ Impossible([Π]))›
4411      using that "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF contingent_denotes[OF that], THEN "≡E"(1)] by blast
4412  next
4413    AOT_assume ‹NonContingent([Π])›
4414    AOT_thus ‹Necessary([Π]) ∨ Impossible([Π])›
4415      using "contingent-properties:3"[THEN "≡dfE"] by blast
4416  qed
4417
4418  show ?thesis
4419  proof (rule "&I")+
4420    AOT_show ‹L ≠ [L]-›
4421      apply (rule "=dfI"(2)[OF L_def])
4422       apply "cqt:2[lambda]"
4423      apply (rule "∀E"(1)[where φ="λ Π . «Π ≠ [Π]-»"])
4424       apply (rule GEN) apply (fact AOT)
4425      by "cqt:2[lambda]"
4426  next
4427    AOT_show ‹L ≠ E!›
4428      apply (rule noneqI)
4429      using "thm-noncont-e-e:3" not_noncontingent_if_contingent[OF "thm-cont-e:5"]
4430      by auto
4431  next
4432    AOT_show ‹L ≠ E!-›
4433      apply (rule noneqI)
4434      using "thm-noncont-e-e:3" apply fast
4435      apply (rule not_noncontingent_if_contingent)
4436      apply (rule "∀E"(1)[where φ="λ Π . «Contingent([Π]) ≡ Contingent([Π]-)»", rotated, OF contingent_denotes, THEN "≡E"(1), rotated])
4437      using "thm-cont-prop:3" GEN apply fast
4438      using "thm-cont-e:5" by fast+
4439  next
4440    AOT_show ‹[L]- ≠ E!-›
4441      apply (rule noneqI)
4442      using "thm-noncont-e-e:4" apply fast
4443      apply (rule not_noncontingent_if_contingent)
4444      apply (rule "∀E"(1)[where φ="λ Π . «Contingent([Π]) ≡ Contingent([Π]-)»", rotated, OF contingent_denotes, THEN "≡E"(1), rotated])
4445      using "thm-cont-prop:3" GEN apply fast
4446      using "thm-cont-e:5" by fast+
4447  next
4448    AOT_show ‹E! ≠ E!-›
4449      apply (rule "=dfI"(2)[OF L_def])
4450       apply "cqt:2[lambda]"
4451      apply (rule "∀E"(1)[where φ="λ Π . «Π ≠ [Π]-»"])
4452       apply (rule GEN) apply (fact AOT)
4453      by (fact "cqt:2[concrete]"[axiom_inst])
4454  qed
4455qed
4456
4457AOT_theorem "thm-cont-propos:1": ‹NonContingent0(p) ≡ NonContingent0(((p)-))›
4458proof(rule "≡I"; rule "→I")
4459  AOT_assume ‹NonContingent0(p)›
4460  AOT_hence ‹Necessary0(p) ∨ Impossible0(p)›
4461    using "contingent-properties:3[zero]"[THEN "≡dfE"] by blast
4462  moreover {
4463    AOT_assume ‹Necessary0(p)›
4464    AOT_hence 1: ‹□p› using "contingent-properties:1[zero]"[THEN "≡dfE"] by blast
4465    AOT_have ‹□¬((p)-)›
4466      by (AOT_subst "«¬((p)-)»" "AOT_term_of_var p")
4467         (auto simp add: 1 "thm-relation-negation:4")
4468    AOT_hence ‹Impossible0(((p)-))›
4469      by (rule "contingent-properties:2[zero]"[THEN "≡dfI"])
4470  }
4471  moreover {
4472    AOT_assume ‹Impossible0(p)›
4473    AOT_hence 1: ‹□¬p›
4474      by (rule "contingent-properties:2[zero]"[THEN "≡dfE"])
4475    AOT_have ‹□((p)-)›
4476      by (AOT_subst "«((p)-)»" "«¬p»") 
4477         (auto simp: 1 "thm-relation-negation:3")
4478    AOT_hence ‹Necessary0(((p)-))›
4479      by (rule "contingent-properties:1[zero]"[THEN "≡dfI"])
4480  }
4481  ultimately AOT_have ‹Necessary0(((p)-)) ∨ Impossible0(((p)-))›
4482    using "∨E"(1) "∨I" "→I" by metis
4483  AOT_thus ‹NonContingent0(((p)-))›
4484    using "contingent-properties:3[zero]"[THEN "≡dfI"] by blast
4485next
4486  AOT_assume ‹NonContingent0(((p)-))›
4487  AOT_hence ‹Necessary0(((p)-)) ∨ Impossible0(((p)-))›
4488    using "contingent-properties:3[zero]"[THEN "≡dfE"] by blast
4489  moreover {
4490    AOT_assume ‹Impossible0(((p)-))›
4491    AOT_hence 1: ‹□¬((p)-)›
4492      by (rule "contingent-properties:2[zero]"[THEN "≡dfE"])
4493    AOT_have ‹□p›
4494      by (AOT_subst_rev "«¬((p)-)»" "AOT_term_of_var p")
4495         (auto simp: 1 "thm-relation-negation:4")
4496    AOT_hence ‹Necessary0(p)›
4497      using "contingent-properties:1[zero]"[THEN "≡dfI"] by blast
4498  }
4499  moreover {
4500    AOT_assume ‹Necessary0(((p)-))›
4501    AOT_hence 1: ‹□((p)-)›
4502      by (rule "contingent-properties:1[zero]"[THEN "≡dfE"])
4503    AOT_have ‹□¬p›
4504      by (AOT_subst_rev "«((p)-)»" "«¬p»")
4505         (auto simp: 1 "thm-relation-negation:3")
4506    AOT_hence ‹Impossible0(p)›
4507      by (rule "contingent-properties:2[zero]"[THEN "≡dfI"])
4508  }
4509  ultimately AOT_have ‹Necessary0(p) ∨ Impossible0(p)›
4510    using "∨E"(1) "∨I" "→I" by metis
4511  AOT_thus ‹NonContingent0(p)›
4512    using "contingent-properties:3[zero]"[THEN "≡dfI"] by blast
4513qed
4514
4515AOT_theorem "thm-cont-propos:2": ‹Contingent0(φ) ≡ ◇φ & ◇¬φ›
4516proof -
4517  AOT_have ‹Contingent0(φ) ≡ ¬(Necessary0(φ) ∨ Impossible0(φ))›
4518    using "contingent-properties:4[zero]"[THEN "≡Df"] by simp
4519  also AOT_have ‹… ≡ ¬Necessary0(φ) & ¬Impossible0(φ)›
4520    by (fact AOT)
4521  also AOT_have ‹… ≡ ¬Impossible0(φ) & ¬Necessary0(φ)›
4522    by (fact AOT)
4523  also AOT_have ‹… ≡ ◇φ & ◇¬φ›
4524    apply (AOT_subst "«◇φ»" "«¬□¬φ»")
4525     apply (simp add: "conventions:5" "≡Df")
4526    apply (AOT_subst "«Impossible0(φ)»" "«□¬φ»")
4527     apply (simp add: "contingent-properties:2[zero]" "≡Df")
4528    apply (AOT_subst_rev "«¬□φ»" "«◇¬φ»")
4529     apply (simp add: "KBasic:11")
4530    apply (AOT_subst "«Necessary0(φ)»" "«□φ»")
4531     apply (simp add: "contingent-properties:1[zero]" "≡Df")
4532    by (simp add: "oth-class-taut:3:a")
4533  finally show ?thesis.
4534qed
4535
4536AOT_theorem "thm-cont-propos:3": ‹Contingent0(p) ≡ Contingent0(((p)-))›
4537proof -
4538  AOT_have ‹Contingent0(p) ≡ ◇p & ◇¬p› using "thm-cont-propos:2".
4539  also AOT_have ‹… ≡ ◇¬p & ◇p› by (fact AOT)
4540  also AOT_have ‹… ≡ ◇((p)-) & ◇p›
4541    by (AOT_subst "«((p)-)»" "«¬p»")
4542       (auto simp: "thm-relation-negation:3" "oth-class-taut:3:a")
4543  also AOT_have ‹… ≡ ◇((p)-) & ◇¬((p)-)›
4544    by (AOT_subst "«¬((p)-)»" "AOT_term_of_var p")
4545       (auto simp: "thm-relation-negation:4" "oth-class-taut:3:a")
4546  also AOT_have ‹… ≡ Contingent0(((p)-))›
4547    using "thm-cont-propos:2"[symmetric] by blast
4548  finally show ?thesis.
4549qed
4550
4551AOT_define noncontingent_prop :: ‹φ› ("p0")
4552  p0_def: "(p0) =df (∀x (E!x → E!x))"
4553
4554AOT_theorem "thm-noncont-propos:1":  ‹Necessary0((p0))›
4555proof(rule "contingent-properties:1[zero]"[THEN "≡dfI"])
4556  AOT_show ‹□(p0)›
4557    apply (rule "=dfI"(2)[OF p0_def])
4558    using "log-prop-prop:2" apply simp
4559    using "if-p-then-p" RN GEN by fast
4560qed
4561
4562AOT_theorem "thm-noncont-propos:2": ‹Impossible0(((p0)-))›
4563proof(rule "contingent-properties:2[zero]"[THEN "≡dfI"])
4564  AOT_show ‹□¬((p0)-)›
4565    apply (AOT_subst "«((p0)-)»" "«¬p0»")
4566    using "thm-relation-negation:3" GEN "∀E"(1)[rotated, OF "log-prop-prop:2"] apply fast
4567    apply (AOT_subst_rev "«p0»" "«¬¬p0»" )
4568     apply (simp add: "oth-class-taut:3:b")
4569    apply (rule "=dfI"(2)[OF p0_def])
4570    using "log-prop-prop:2" apply simp
4571    using "if-p-then-p" RN GEN by fast
4572qed
4573
4574AOT_theorem "thm-noncont-propos:3": ‹NonContingent0((p0))›
4575  apply(rule "contingent-properties:3[zero]"[THEN "≡dfI"])
4576  using "thm-noncont-propos:1" "∨I" by blast
4577
4578AOT_theorem "thm-noncont-propos:4": ‹NonContingent0(((p0)-))›
4579  apply(rule "contingent-properties:3[zero]"[THEN "≡dfI"])
4580  using "thm-noncont-propos:2" "∨I" by blast
4581
4582AOT_theorem "thm-noncont-propos:5": ‹∃p∃q (NonContingent0((p)) & NonContingent0((q)) & p ≠ q)›
4583proof(rule "∃I")+
4584  AOT_have 0: ‹φ ≠ (φ)-› for φ
4585    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4586  AOT_thus ‹NonContingent0((p0)) & NonContingent0(((p0)-)) & (p0) ≠ (p0)-›
4587    using "thm-noncont-propos:3" "thm-noncont-propos:4" "&I" by auto
4588qed(auto simp: "log-prop-prop:2")
4589
4590AOT_act_theorem "no-cnac": ‹¬∃x(E!x & ¬𝒜E!x)›
4591proof(rule "raa-cor:2")
4592  AOT_assume ‹∃x(E!x & ¬𝒜E!x)›
4593  then AOT_obtain a where a: ‹E!a & ¬𝒜E!a›
4594    using "∃E"[rotated] by blast
4595  AOT_hence ‹𝒜¬E!a› using "&E" "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
4596  AOT_hence ‹¬E!a› using "logic-actual"[act_axiom_inst, THEN "→E"] by blast
4597  AOT_hence ‹E!a & ¬E!a› using a "&E" "&I" by blast
4598  AOT_thus ‹p & ¬p› for p using "raa-cor:1" by blast
4599qed
4600
4601AOT_theorem "pos-not-pna:1": ‹¬𝒜∃x (E!x & ¬𝒜E!x)›
4602proof(rule "raa-cor:2")
4603  AOT_assume ‹𝒜∃x (E!x & ¬𝒜E!x)›
4604  AOT_hence ‹∃x 𝒜(E!x & ¬𝒜E!x)›
4605    using "Act-Basic:10"[THEN "≡E"(1)] by blast
4606  then AOT_obtain a where ‹𝒜(E!a & ¬𝒜E!a)› using "∃E"[rotated] by blast
4607  AOT_hence 1: ‹𝒜E!a & 𝒜¬𝒜E!a› using "Act-Basic:2"[THEN "≡E"(1)] by blast
4608  AOT_hence ‹¬𝒜𝒜E!a› using "&E"(2) "logic-actual-nec:1"[axiom_inst, THEN "≡E"(1)] by blast
4609  AOT_hence ‹¬𝒜E!a› using "logic-actual-nec:4"[axiom_inst, THEN "≡E"(1)] RAA by blast
4610  AOT_thus ‹p & ¬p› for p using 1[THEN "&E"(1)] "&I" "raa-cor:1" by blast
4611qed
4612
4613AOT_theorem "pos-not-pna:2": ‹◇¬∃x(E!x & ¬𝒜E!x)›
4614proof (rule RAA(1))
4615  AOT_show ‹¬𝒜∃x (E!x & ¬𝒜E!x)› using "pos-not-pna:1" by blast
4616next
4617  AOT_assume ‹¬◇¬∃x (E!x & ¬𝒜E!x)›
4618  AOT_hence ‹□∃x (E!x & ¬𝒜E!x)›
4619    using "KBasic:12"[THEN "≡E"(2)] by blast
4620  AOT_thus ‹𝒜∃x (E!x & ¬𝒜E!x)›
4621    using "nec-imp-act"[THEN "→E"] by blast
4622qed
4623
4624AOT_theorem "pos-not-pna:3": ‹∃x (◇E!x & ¬𝒜E!x)›
4625proof -
4626  AOT_obtain a where ‹◇(E!a & ¬𝒜E!a)›
4627    using "qml:4"[axiom_inst] "BF◇"[THEN "→E"] "∃E"[rotated] by blast
4628  AOT_hence θ: ‹◇E!a› and ξ: ‹◇¬𝒜E!a› using "KBasic2:3"[THEN "→E"] "&E" by blast+
4629  AOT_have ‹¬□𝒜E!a› using ξ "KBasic:11"[THEN "≡E"(2)] by blast
4630  AOT_hence ‹¬𝒜E!a› using "Act-Basic:6"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)] by blast
4631  AOT_hence ‹◇E!a & ¬𝒜E!a› using θ "&I" by blast
4632  thus ?thesis using "∃I" by fast
4633qed
4634
4635AOT_define contingent_prop :: φ ("q0")
4636  q0_def: ‹(q0) =df (∃x (E!x & ¬𝒜E!x))›
4637
4638AOT_theorem q0_prop: ‹◇q0 & ◇¬q0›
4639  apply (rule "=dfI"(2)[OF q0_def])
4640  apply (fact "log-prop-prop:2")
4641  apply (rule "&I")
4642   apply (fact "qml:4"[axiom_inst])
4643  by (fact "pos-not-pna:2")
4644
4645AOT_theorem "basic-prop:1": ‹Contingent0((q0))›
4646proof(rule "contingent-properties:4[zero]"[THEN "≡dfI"])
4647  AOT_have ‹¬Necessary0((q0)) & ¬Impossible0((q0))›
4648  proof (rule "&I"; rule "=dfI"(2)[OF q0_def]; (rule "log-prop-prop:2" | rule "raa-cor:2"))
4649    AOT_assume ‹Necessary0(∃x (E!x & ¬𝒜E!x))›
4650    AOT_hence ‹□∃x (E!x & ¬𝒜E!x)›
4651      using "contingent-properties:1[zero]"[THEN "≡dfE"] by blast
4652    AOT_hence ‹𝒜∃x (E!x & ¬𝒜E!x)›
4653      using "Act-Basic:8"[THEN "→E"] "qml:2"[axiom_inst, THEN "→E"] by blast
4654    AOT_thus ‹𝒜∃x (E!x & ¬𝒜E!x) & ¬𝒜∃x (E!x & ¬𝒜E!x)›
4655      using "pos-not-pna:1" "&I" by blast
4656  next
4657    AOT_assume ‹Impossible0(∃x (E!x & ¬𝒜E!x))›
4658    AOT_hence ‹□¬(∃x (E!x & ¬𝒜E!x))›
4659      using "contingent-properties:2[zero]"[THEN "≡dfE"] by blast
4660    AOT_hence ‹¬◇(∃x (E!x & ¬𝒜E!x))› using "KBasic2:1"[THEN "≡E"(1)] by blast
4661    AOT_thus ‹◇(∃x (E!x & ¬𝒜E!x)) & ¬◇(∃x (E!x & ¬𝒜E!x))›
4662      using "qml:4"[axiom_inst] "&I" by blast
4663  qed
4664  AOT_thus ‹¬(Necessary0((q0)) ∨ Impossible0((q0)))›
4665    using "oth-class-taut:5:d" "≡E"(2) by blast
4666qed
4667
4668AOT_theorem "basic-prop:2": ‹∃p Contingent0((p))›
4669  using "∃I"(1)[rotated, OF "log-prop-prop:2"] "basic-prop:1" by blast
4670
4671AOT_theorem "basic-prop:3": ‹Contingent0(((q0)-))›
4672  apply (AOT_subst "«(q0)-»" "«¬q0»")
4673   apply (insert "thm-relation-negation:3" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"]; fast)
4674  apply (rule "contingent-properties:4[zero]"[THEN "≡dfI"])
4675  apply (rule "oth-class-taut:5:d"[THEN "≡E"(2)])
4676  apply (rule "&I")
4677   apply (rule "contingent-properties:1[zero]"[THEN "df-rules-formulas[3]", THEN "useful-tautologies:5"[THEN "→E"], THEN "→E"])
4678   apply (rule "conventions:5"[THEN "≡dfE"])
4679   apply (rule "=dfE"(2)[OF q0_def])
4680    apply (rule "log-prop-prop:2")
4681   apply (rule q0_prop[THEN "&E"(1)])
4682  apply (rule "contingent-properties:2[zero]"[THEN "df-rules-formulas[3]", THEN "useful-tautologies:5"[THEN "→E"], THEN "→E"])
4683  apply (rule "conventions:5"[THEN "≡dfE"])
4684  by (rule q0_prop[THEN "&E"(2)])
4685
4686AOT_theorem "basic-prop:4": ‹∃p∃q (p ≠ q & Contingent0(p) & Contingent0(q))›
4687proof(rule "∃I")+
4688  AOT_have 0: ‹φ ≠ (φ)-› for φ
4689    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4690  AOT_show ‹(q0) ≠ (q0)- & Contingent0(q0) & Contingent0(((q0)-))›
4691    using "basic-prop:1" "basic-prop:3" "&I" 0 by presburger
4692qed(auto simp: "log-prop-prop:2")
4693
4694AOT_theorem "proposition-facts:1": ‹NonContingent0(p) → ¬∃q (Contingent0(q) & q = p)›
4695proof(rule "→I"; rule "raa-cor:2")
4696  AOT_assume ‹NonContingent0(p)›
4697  AOT_hence 1: ‹Necessary0(p) ∨ Impossible0(p)›
4698    using "contingent-properties:3[zero]"[THEN "≡dfE"] by blast
4699  AOT_assume ‹∃q (Contingent0(q) & q = p)›
4700  then AOT_obtain q where ‹Contingent0(q) & q = p› using "∃E"[rotated] by blast
4701  AOT_hence ‹Contingent0(p)› using "rule=E" "&E" by fast
4702  AOT_thus ‹(Necessary0(p) ∨ Impossible0(p)) & ¬(Necessary0(p) ∨ Impossible0(p))›
4703    using "contingent-properties:4[zero]"[THEN "≡dfE"] 1 "&I" by blast
4704qed
4705
4706AOT_theorem "proposition-facts:2": ‹Contingent0(p) → ¬∃q (NonContingent0(q) & q = p)›
4707proof(rule "→I"; rule "raa-cor:2")
4708  AOT_assume ‹Contingent0(p)›
4709  AOT_hence 1: ‹¬(Necessary0(p) ∨ Impossible0(p))›
4710    using "contingent-properties:4[zero]"[THEN "≡dfE"] by blast
4711  AOT_assume ‹∃q (NonContingent0(q) & q = p)›
4712  then AOT_obtain q where ‹NonContingent0(q) & q = p› using "∃E"[rotated] by blast
4713  AOT_hence ‹NonContingent0(p)› using "rule=E" "&E" by fast
4714  AOT_thus ‹(Necessary0(p) ∨ Impossible0(p)) & ¬(Necessary0(p) ∨ Impossible0(p))›
4715    using "contingent-properties:3[zero]"[THEN "≡dfE"] 1 "&I" by blast
4716qed
4717
4718AOT_theorem "proposition-facts:3": ‹(p0) ≠ (p0)- & (p0) ≠ (q0) & (p0) ≠ (q0)- & (p0)- ≠ (q0)- & (q0) ≠ (q0)-›
4719proof -
4720  {
4721    fix χ φ ψ
4722    AOT_assume ‹χ{φ}›
4723    moreover AOT_assume ‹¬χ{ψ}›
4724    ultimately AOT_have ‹¬(χ{φ} ≡ χ{ψ})›
4725      using RAA "≡E" by metis
4726    moreover {
4727      AOT_have ‹∀p∀q ((¬(χ{p} ≡ χ{q})) → p ≠ q)›
4728        by (rule "∀I"; rule "∀I"; rule "pos-not-equiv-ne:4[zero]")
4729      AOT_hence ‹((¬(χ{φ} ≡ χ{ψ})) → φ ≠ ψ)›
4730        using "∀E" "log-prop-prop:2" by blast
4731    }
4732    ultimately AOT_have ‹φ ≠ ψ›
4733      using "→E" by blast
4734  } note 0 = this
4735  AOT_have contingent_neg: ‹Contingent0(φ) ≡ Contingent0(((φ)-))› for φ
4736    using "thm-cont-propos:3" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4737  AOT_have not_noncontingent_if_contingent: ‹¬NonContingent0(φ)› if ‹Contingent0(φ)› for φ
4738    apply (rule "contingent-properties:3[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4739    using that "contingent-properties:4[zero]"[THEN "≡dfE"] by blast
4740  show ?thesis
4741    apply (rule "&I")+
4742    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] apply fast
4743       apply (rule 0)
4744    using "thm-noncont-propos:3" apply fast
4745       apply (rule not_noncontingent_if_contingent)
4746       apply (fact AOT)
4747      apply (rule 0)
4748    apply (rule "thm-noncont-propos:3")
4749      apply (rule not_noncontingent_if_contingent)
4750      apply (rule contingent_neg[THEN "≡E"(1)])
4751      apply (fact AOT)
4752     apply (rule 0)
4753    apply (rule "thm-noncont-propos:4")
4754      apply (rule not_noncontingent_if_contingent)
4755      apply (rule contingent_neg[THEN "≡E"(1)])
4756     apply (fact AOT)
4757    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4758qed
4759
4760AOT_define "cont-tf:1" :: ‹φ ⇒ φ› ("ContingentlyTrue'(_')")
4761  "cont-tf:1": ‹ContingentlyTrue(p) ≡df p & ◇¬p›
4762
4763AOT_define "cont-tf:2" :: ‹φ ⇒ φ› ("ContingentlyFalse'(_')")
4764  "cont-tf:2": ‹ContingentlyFalse(p) ≡df ¬p & ◇p›
4765
4766AOT_theorem "cont-true-cont:1": ‹ContingentlyTrue((p)) → Contingent0((p))›
4767proof(rule "→I")
4768  AOT_assume ‹ContingentlyTrue((p))›
4769  AOT_hence 1: ‹p› and 2: ‹◇¬p› using "cont-tf:1"[THEN "≡dfE"] "&E" by blast+
4770  AOT_have ‹¬Necessary0((p))›
4771    apply (rule "contingent-properties:1[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4772    using 2 "KBasic:11"[THEN "≡E"(2)] by blast
4773  moreover AOT_have ‹¬Impossible0((p))›
4774    apply (rule "contingent-properties:2[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4775    apply (rule "conventions:5"[THEN "≡dfE"])
4776    using "T◇"[THEN "→E", OF 1].
4777  ultimately AOT_have ‹¬(Necessary0((p)) ∨ Impossible0((p)))›
4778    using DeMorgan(2)[THEN "≡E"(2)] "&I" by blast
4779  AOT_thus ‹Contingent0((p))›
4780    using "contingent-properties:4[zero]"[THEN "≡dfI"] by blast
4781qed
4782
4783AOT_theorem "cont-true-cont:2": ‹ContingentlyFalse((p)) → Contingent0((p))›
4784proof(rule "→I")
4785  AOT_assume ‹ContingentlyFalse((p))›
4786  AOT_hence 1: ‹¬p› and 2: ‹◇p› using "cont-tf:2"[THEN "≡dfE"] "&E" by blast+
4787  AOT_have ‹¬Necessary0((p))›
4788    apply (rule "contingent-properties:1[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4789    using "KBasic:11"[THEN "≡E"(2)] "T◇"[THEN "→E", OF 1] by blast
4790  moreover AOT_have ‹¬Impossible0((p))›
4791    apply (rule "contingent-properties:2[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4792    apply (rule "conventions:5"[THEN "≡dfE"])
4793    using 2.
4794  ultimately AOT_have ‹¬(Necessary0((p)) ∨ Impossible0((p)))›
4795    using DeMorgan(2)[THEN "≡E"(2)] "&I" by blast
4796  AOT_thus ‹Contingent0((p))›
4797    using "contingent-properties:4[zero]"[THEN "≡dfI"] by blast
4798qed
4799
4800AOT_theorem "cont-true-cont:3": ‹ContingentlyTrue((p)) ≡ ContingentlyFalse(((p)-))›
4801proof(rule "≡I"; rule "→I")
4802  AOT_assume ‹ContingentlyTrue((p))›
4803  AOT_hence 0: ‹p & ◇¬p› using "cont-tf:1"[THEN "≡dfE"] by blast
4804  AOT_have 1: ‹ContingentlyFalse(¬p)›
4805    apply (rule "cont-tf:2"[THEN "≡dfI"])
4806    apply (AOT_subst_rev "AOT_term_of_var p" "«¬¬p»")
4807    by (auto simp: "oth-class-taut:3:b" 0)
4808  AOT_show ‹ContingentlyFalse(((p)-))›
4809    apply (AOT_subst "«(p)-»" "«¬p»")
4810    by (auto simp: "thm-relation-negation:3" 1)
4811next
4812  AOT_assume 1: ‹ContingentlyFalse(((p)-))›
4813  AOT_have ‹ContingentlyFalse(¬p)›
4814    by (AOT_subst_rev "«(p)-»" "«¬p»")
4815       (auto simp: "thm-relation-negation:3" 1)
4816  AOT_hence ‹¬¬p & ◇¬p› using "cont-tf:2"[THEN "≡dfE"] by blast
4817  AOT_hence ‹p & ◇¬p›
4818    using "&I" "&E" "useful-tautologies:1"[THEN "→E"] by metis
4819  AOT_thus ‹ContingentlyTrue((p))›
4820    using "cont-tf:1"[THEN "≡dfI"] by blast
4821qed
4822
4823AOT_theorem "cont-true-cont:4": ‹ContingentlyFalse((p)) ≡ ContingentlyTrue(((p)-))›
4824proof(rule "≡I"; rule "→I")
4825  AOT_assume ‹ContingentlyFalse(p)›
4826  AOT_hence 0: ‹¬p & ◇p›
4827    using "cont-tf:2"[THEN "≡dfE"] by blast
4828  AOT_have ‹¬p & ◇¬¬p›
4829    by (AOT_subst_rev "AOT_term_of_var p" "«¬¬p»")
4830       (auto simp: "oth-class-taut:3:b" 0)
4831  AOT_hence 1: ‹ContingentlyTrue(¬p)›
4832    by (rule "cont-tf:1"[THEN "≡dfI"])
4833  AOT_show ‹ContingentlyTrue(((p)-))›
4834    by (AOT_subst "«(p)-»" "«¬p»")
4835       (auto simp: "thm-relation-negation:3" 1)
4836next
4837  AOT_assume 1: ‹ContingentlyTrue(((p)-))›
4838  AOT_have ‹ContingentlyTrue(¬p)›
4839    by (AOT_subst_rev "«(p)-»" "«¬p»")
4840       (auto simp add: "thm-relation-negation:3" 1)
4841  AOT_hence 2: ‹¬p & ◇¬¬p› using "cont-tf:1"[THEN "≡dfE"] by blast
4842  AOT_have ‹◇p›
4843    by (AOT_subst "AOT_term_of_var p" "«¬¬p»")
4844       (auto simp add: "oth-class-taut:3:b" 2[THEN "&E"(2)])
4845  AOT_hence ‹¬p & ◇p› using 2[THEN "&E"(1)] "&I" by blast
4846  AOT_thus ‹ContingentlyFalse(p)›
4847    by (rule "cont-tf:2"[THEN "≡dfI"])
4848qed
4849
4850AOT_theorem "cont-true-cont:5": ‹(ContingentlyTrue((p)) & Necessary0((q))) → p ≠ q›
4851proof (rule "→I"; frule "&E"(1); drule "&E"(2); rule "raa-cor:1")
4852  AOT_assume ‹ContingentlyTrue((p))›
4853  AOT_hence ‹◇¬p›
4854    using "cont-tf:1"[THEN "≡dfE"] "&E" by blast
4855  AOT_hence 0: ‹¬□p› using "KBasic:11"[THEN "≡E"(2)] by blast
4856  AOT_assume ‹Necessary0((q))›
4857  moreover AOT_assume ‹¬(p ≠ q)›
4858  AOT_hence ‹p = q›
4859    using "=-infix"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4860          "useful-tautologies:1"[THEN "→E"] by blast
4861  ultimately AOT_have ‹Necessary0((p))› using "rule=E" id_sym by blast
4862  AOT_hence ‹□p›
4863    using "contingent-properties:1[zero]"[THEN "≡dfE"] by blast
4864  AOT_thus ‹□p & ¬□p› using 0 "&I" by blast
4865qed
4866
4867AOT_theorem "cont-true-cont:6": ‹(ContingentlyFalse((p)) & Impossible0((q))) → p ≠ q›
4868proof (rule "→I"; frule "&E"(1); drule "&E"(2); rule "raa-cor:1")
4869  AOT_assume ‹ContingentlyFalse((p))›
4870  AOT_hence ‹◇p›
4871    using "cont-tf:2"[THEN "≡dfE"] "&E" by blast
4872  AOT_hence 1: ‹¬□¬p›
4873    using "conventions:5"[THEN "≡dfE"] by blast
4874  AOT_assume ‹Impossible0((q))›
4875  moreover AOT_assume ‹¬(p ≠ q)›
4876  AOT_hence ‹p = q›
4877    using "=-infix"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4878          "useful-tautologies:1"[THEN "→E"] by blast
4879  ultimately AOT_have ‹Impossible0((p))› using "rule=E" id_sym by blast
4880  AOT_hence ‹□¬p›
4881    using "contingent-properties:2[zero]"[THEN "≡dfE"] by blast
4882  AOT_thus ‹□¬p & ¬□¬p› using 1 "&I" by blast
4883qed
4884
4885AOT_act_theorem "q0cf:1": ‹ContingentlyFalse(q0)›
4886  apply (rule "cont-tf:2"[THEN "≡dfI"])
4887  apply (rule "=dfI"(2)[OF q0_def])
4888   apply (fact "log-prop-prop:2")
4889  apply (rule "&I")
4890   apply (fact "no-cnac")
4891  by (fact "qml:4"[axiom_inst])
4892
4893AOT_act_theorem "q0cf:2": ‹ContingentlyTrue(((q0)-))›
4894  apply (rule "cont-tf:1"[THEN "≡dfI"])
4895  apply (rule "=dfI"(2)[OF q0_def])
4896   apply (fact "log-prop-prop:2")
4897  apply (rule "&I")
4898     apply (rule "thm-relation-negation:3"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(2)])
4899     apply (fact "no-cnac")
4900    apply (rule "rule=E"[rotated, OF "thm-relation-negation:7"[unvarify p, OF "log-prop-prop:2", THEN id_sym]])
4901  apply (AOT_subst_rev "«∃x (E!x & ¬𝒜E!x)»" "«¬¬(∃x  (E!x & ¬𝒜E!x))»")
4902  by (auto simp: "oth-class-taut:3:b" "qml:4"[axiom_inst])
4903
4904(* TODO: q0cf-rem skipped for now *)
4905
4906AOT_theorem "cont-tf-thm:1": ‹∃p ContingentlyTrue((p))›
4907proof(rule "∨E"(1)[OF "exc-mid"]; rule "→I"; rule "∃I")
4908  AOT_assume ‹q0›
4909  AOT_hence ‹q0 & ◇¬q0› using q0_prop[THEN "&E"(2)] "&I" by blast
4910  AOT_thus ‹ContingentlyTrue(q0)›
4911    by (rule "cont-tf:1"[THEN "≡dfI"])
4912next
4913  AOT_assume ‹¬q0›
4914  AOT_hence ‹¬q0 & ◇q0› using q0_prop[THEN "&E"(1)] "&I" by blast
4915  AOT_hence ‹ContingentlyFalse(q0)›
4916    by (rule "cont-tf:2"[THEN "≡dfI"])
4917  AOT_thus ‹ContingentlyTrue(((q0)-))›
4918    by (rule "cont-true-cont:4"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(1)])
4919qed(auto simp: "log-prop-prop:2")
4920
4921
4922AOT_theorem "cont-tf-thm:2": ‹∃p ContingentlyFalse((p))›
4923proof(rule "∨E"(1)[OF "exc-mid"]; rule "→I"; rule "∃I")
4924  AOT_assume ‹q0›
4925  AOT_hence ‹q0 & ◇¬q0› using q0_prop[THEN "&E"(2)] "&I" by blast
4926  AOT_hence ‹ContingentlyTrue(q0)›
4927    by (rule "cont-tf:1"[THEN "≡dfI"])
4928  AOT_thus ‹ContingentlyFalse(((q0)-))›
4929    by (rule "cont-true-cont:3"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(1)])
4930next
4931  AOT_assume ‹¬q0›
4932  AOT_hence ‹¬q0 & ◇q0› using q0_prop[THEN "&E"(1)] "&I" by blast
4933  AOT_thus ‹ContingentlyFalse(q0)›
4934    by (rule "cont-tf:2"[THEN "≡dfI"])
4935qed(auto simp: "log-prop-prop:2")
4936
4937(* TODO: inspect modally strict subproof involving obtained variable *)
4938AOT_theorem "property-facts1:1": ‹∃F∃x ([F]x & ◇¬[F]x)›
4939proof -
4940  fix x
4941  AOT_obtain p1 where ‹ContingentlyTrue((p1))›
4942    using "cont-tf-thm:1" "∃E"[rotated] by blast
4943  AOT_hence 1: ‹p1 & ◇¬p1› using "cont-tf:1"[THEN "≡dfE"] by blast
4944  AOT_modally_strict {
4945    AOT_have ‹for arbitrary p: ⊢□ ([λz p]x ≡ p)›
4946      by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
4947    AOT_hence ‹for arbitrary p: ⊢□ □ ([λz p]x ≡ p)›
4948      by (rule RN)
4949    AOT_hence ‹∀p □([λz p]x ≡ p)› using GEN by fast
4950    AOT_hence ‹□([λz p1]x ≡ p1)› using "∀E" by fast
4951  } note 2 = this
4952  AOT_hence ‹□([λz p1]x ≡ p1)› using "∀E" by blast
4953  AOT_hence ‹[λz p1]x› using 1[THEN "&E"(1)] "qml:2"[axiom_inst, THEN "→E"] "≡E"(2) by blast
4954  moreover AOT_have ‹◇¬[λz p1]x›
4955    apply (AOT_subst_using subst: 2[THEN "qml:2"[axiom_inst, THEN "→E"]])
4956    using 1[THEN "&E"(2)] by blast
4957  ultimately AOT_have ‹[λz p1]x & ◇¬[λz p1]x› using "&I" by blast
4958  AOT_hence ‹∃x ([λz p1]x & ◇¬[λz p1]x)› using "∃I"(2) by fast
4959  moreover AOT_have ‹[λz p1]↓› by "cqt:2[lambda]"
4960  ultimately AOT_show ‹∃F∃x ([F]x & ◇¬[F]x)› by (rule "∃I"(1))
4961qed
4962
4963(* TODO: inspect modally strict subproof involving obtained variable *)
4964AOT_theorem "property-facts1:2": ‹∃F∃x (¬[F]x & ◇[F]x)›
4965proof -
4966  fix x
4967  AOT_obtain p1 where ‹ContingentlyFalse((p1))›
4968    using "cont-tf-thm:2" "∃E"[rotated] by blast
4969  AOT_hence 1: ‹¬p1 & ◇p1› using "cont-tf:2"[THEN "≡dfE"] by blast
4970  AOT_modally_strict {
4971    AOT_have ‹for arbitrary p: ⊢□ ([λz p]x ≡ p)›
4972      by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
4973    AOT_hence ‹for arbitrary p: ⊢□ (¬[λz p]x ≡ ¬p)›
4974      using "oth-class-taut:4:b" "≡E" by blast
4975    AOT_hence ‹for arbitrary p: ⊢□ □(¬[λz p]x ≡ ¬p)›
4976      by (rule RN)
4977    AOT_hence ‹∀p □(¬[λz p]x ≡ ¬p)› using GEN by fast
4978    AOT_hence ‹□(¬[λz p1]x ≡ ¬p1)› using "∀E" by fast
4979  } note 2 = this
4980  AOT_hence ‹□(¬[λz p1]x ≡ ¬p1)› using "∀E" by blast
4981  AOT_hence 3: ‹¬[λz p1]x› using 1[THEN "&E"(1)] "qml:2"[axiom_inst, THEN "→E"] "≡E"(2) by blast
4982  AOT_modally_strict {
4983    AOT_have ‹for arbitrary p: ⊢□ ([λz p]x ≡ p)›
4984      by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
4985    AOT_hence ‹for arbitrary p: ⊢□ □([λz p]x ≡ p)›
4986      by (rule RN)
4987    AOT_hence ‹∀p □([λz p]x ≡ p)› using GEN by fast
4988    AOT_hence ‹□([λz p1]x ≡ p1)› using "∀E" by fast
4989  } note 4 = this
4990  AOT_have ‹◇[λz p1]x›
4991    apply (AOT_subst_using subst: 4[THEN "qml:2"[axiom_inst, THEN "→E"]])
4992    using 1[THEN "&E"(2)] by blast
4993  AOT_hence ‹¬[λz p1]x & ◇[λz p1]x› using 3 "&I" by blast
4994  AOT_hence ‹∃x (¬[λz p1]x & ◇[λz p1]x)› using "∃I"(2) by fast
4995  moreover AOT_have ‹[λz p1]↓› by "cqt:2[lambda]"
4996  ultimately AOT_show ‹∃F∃x (¬[F]x & ◇[F]x)› by (rule "∃I"(1))
4997qed
4998
4999context
5000begin
5001
5002private AOT_lemma eqnotnec_123_Aux_ζ: ‹[L]x ≡ (E!x → E!x)›
5003    apply (rule "=dfI"(2)[OF L_def])
5004     apply "cqt:2[lambda]"
5005    apply (rule "beta-C-meta"[THEN "→E"])
5006  by "cqt:2[lambda]"
5007
5008private AOT_lemma eqnotnec_123_Aux_ω: ‹[λz φ]x ≡ φ›
5009    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5010
5011private AOT_lemma eqnotnec_123_Aux_θ: ‹φ ≡ ∀x([L]x ≡ [λz φ]x)›
5012proof(rule "≡I"; rule "→I"; (rule "∀I")?)
5013  fix x
5014  AOT_assume 1: ‹φ›
5015  AOT_have ‹[L]x ≡ (E!x → E!x)› using eqnotnec_123_Aux_ζ.
5016  also AOT_have ‹… ≡ φ›
5017    using "if-p-then-p" 1 "≡I" "→I" by simp
5018  also AOT_have ‹… ≡ [λz φ]x›
5019    using "Commutativity of ≡"[THEN "≡E"(1)] eqnotnec_123_Aux_ω by blast
5020  finally AOT_show ‹[L]x ≡ [λz φ]x›.
5021next
5022  fix x
5023  AOT_assume ‹∀x([L]x ≡ [λz φ]x)›
5024  AOT_hence ‹[L]x ≡ [λz φ]x› using "∀E" by blast
5025  also AOT_have ‹… ≡ φ› using eqnotnec_123_Aux_ω.
5026  finally AOT_have ‹φ ≡ [L]x› using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5027  also AOT_have ‹… ≡ E!x → E!x› using eqnotnec_123_Aux_ζ.
5028  finally AOT_show ‹φ› using "≡E" "if-p-then-p" by fast
5029qed
5030private lemmas eqnotnec_123_Aux_ξ =  eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
5031                      THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)],
5032                      THEN "RM◇"]
5033private lemmas eqnotnec_123_Aux_ξ' = eqnotnec_123_Aux_θ[THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)], THEN "RM◇"]
5034
5035AOT_theorem "eqnotnec:1": ‹∃F∃G(∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5036proof-
5037  AOT_obtain p1 where ‹ContingentlyTrue(p1)› using "cont-tf-thm:1" "∃E"[rotated] by blast
5038  AOT_hence ‹p1 & ◇¬p1› using "cont-tf:1"[THEN "≡dfE"] by blast
5039  AOT_hence ‹∀x ([L]x ≡ [λz p1]x) & ◇¬∀x([L]x ≡ [λz p1]x)›
5040    apply - apply (rule "&I")
5041    using "&E" eqnotnec_123_Aux_θ[THEN "≡E"(1)] eqnotnec_123_Aux_ξ "→E" by fast+
5042  AOT_hence ‹∃G (∀x([L]x ≡ [G]x) & ◇¬∀x([L]x ≡ [G]x))›
5043    by (rule "∃I") "cqt:2[lambda]"
5044  AOT_thus ‹∃F∃G (∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5045    apply (rule "∃I")
5046    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5047qed
5048
5049AOT_theorem "eqnotnec:2": ‹∃F∃G(¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5050proof-
5051  AOT_obtain p1 where ‹ContingentlyFalse(p1)› using "cont-tf-thm:2" "∃E"[rotated] by blast
5052  AOT_hence ‹¬p1 & ◇p1› using "cont-tf:2"[THEN "≡dfE"] by blast
5053  AOT_hence ‹¬∀x ([L]x ≡ [λz p1]x) & ◇∀x([L]x ≡ [λz p1]x)›
5054    apply - apply (rule "&I")
5055    using "&E" eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)] eqnotnec_123_Aux_ξ' "→E" by fast+
5056  AOT_hence ‹∃G (¬∀x([L]x ≡ [G]x) & ◇∀x([L]x ≡ [G]x))›
5057    by (rule "∃I") "cqt:2[lambda]"
5058  AOT_thus ‹∃F∃G (¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5059    apply (rule "∃I")
5060    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5061qed
5062
5063AOT_theorem "eqnotnec:3": ‹∃F∃G(𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5064proof-
5065  AOT_have ‹¬𝒜q0›
5066    apply (rule "=dfI"(2)[OF q0_def])
5067     apply (fact "log-prop-prop:2")
5068    by (fact AOT)
5069  AOT_hence ‹𝒜¬q0›
5070    using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5071  AOT_hence ‹𝒜¬∀x ([L]x ≡ [λz q0]x)›
5072    using eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
5073            THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)],
5074            THEN "RA[2]", THEN "act-cond"[THEN "→E"], THEN "→E"] by blast
5075  moreover AOT_have ‹◇∀x ([L]x ≡ [λz q0]x)› using eqnotnec_123_Aux_ξ'[THEN "→E"] q0_prop[THEN "&E"(1)] by blast
5076  ultimately AOT_have ‹𝒜¬∀x ([L]x ≡ [λz q0]x) & ◇∀x ([L]x ≡ [λz q0]x)› using "&I" by blast
5077  AOT_hence ‹∃G (𝒜¬∀x([L]x ≡ [G]x) & ◇∀x([L]x ≡ [G]x))›
5078    by (rule "∃I") "cqt:2[lambda]"
5079  AOT_thus ‹∃F∃G (𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5080    apply (rule "∃I")
5081    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5082qed
5083
5084end
5085
5086(* TODO[IMPORTANT]: proof of 219.4 ζ: appeal to (159.2) requires a theorem, but the result has local
5087   assumptions! *)
5088AOT_theorem "eqnotnec:4": ‹∀F∃G(∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5089proof(rule GEN)
5090  fix F
5091
5092  AOT_have Aux_A: ‹⊢□ ψ → ∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5093  proof(rule "→I"; rule GEN)
5094    AOT_modally_strict {
5095    fix x
5096    AOT_assume 0: ‹ψ›
5097    AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5098      by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5099    also AOT_have ‹... ≡ [F]x›
5100      apply (rule "≡I"; rule "→I")
5101      using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5102      using 0 "&I" by blast
5103    finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5104      using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5105    }
5106  qed
5107
5108  AOT_have Aux_B: ‹⊢□ ψ → ∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5109  proof (rule "→I"; rule GEN)
5110    AOT_modally_strict {
5111      fix x
5112      AOT_assume 0: ‹ψ›
5113      AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5114        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5115      also AOT_have ‹... ≡ [F]x›
5116        apply (rule "≡I"; rule "→I")
5117        using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5118        apply (rule "∨I"(1)) using 0 "&I" by blast
5119      finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5120        using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5121    }
5122  qed
5123
5124  AOT_have Aux_C: ‹⊢□ ◇¬ψ → ◇¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5125  proof(rule "RM◇"; rule "→I"; rule "raa-cor:2")
5126  AOT_modally_strict {
5127      AOT_assume 0: ‹¬ψ›
5128      AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5129      AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5130      moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5131          by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5132      moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5133        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5134      ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5135        using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5136      moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5137      ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5138      AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5139    }
5140  qed
5141
5142  AOT_have Aux_D: ‹□∀z ([F]z ≡ [λz [F]z & ψ]z) → (◇¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ◇¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5143  proof (rule "→I")
5144    AOT_assume A: ‹□∀z([F]z ≡ [λz [F]z & ψ]z)›
5145    AOT_show ‹◇¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ◇¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5146    proof(rule "≡I"; rule "KBasic:13"[THEN "→E"];
5147          rule "RN[prem]"[where Γ="{«∀z([F]z ≡ [λz [F]z & ψ]z)»}", simplified];
5148          (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5149      AOT_modally_strict {
5150        AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5151        AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5152        AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5153        AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5154        AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5155        AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5156      }
5157    next
5158      AOT_modally_strict {
5159        AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5160        AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5161        AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5162        AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5163        AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5164        AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5165      }
5166    qed(auto simp: A)
5167  qed
5168
5169  AOT_obtain p1 where p1_prop: ‹p1 & ◇¬p1› using "cont-tf-thm:1" "∃E"[rotated] "cont-tf:1"[THEN "≡dfE"] by blast
5170  {
5171    AOT_assume 1: ‹□∀x([F]x ≡ [λz [F]z & p1]x)›
5172    AOT_have 2: ‹∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5173      using Aux_B[THEN "→E", OF p1_prop[THEN "&E"(1)]].
5174    AOT_have ‹◇¬∀x([λz [F]z & p1]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5175      using Aux_C[THEN "→E", OF p1_prop[THEN "&E"(2)]].
5176    AOT_hence 3: ‹◇¬∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5177      using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5178    AOT_hence ‹∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x) & ◇¬∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)› using 2 "&I" by blast
5179    AOT_hence ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5180      by (rule "∃I"(1)) "cqt:2[lambda]"
5181  }
5182  moreover {
5183    AOT_assume 2: ‹¬□∀x([F]x ≡ [λz [F]z & p1]x)›
5184    AOT_hence ‹◇¬∀x([F]x ≡ [λz [F]z & p1]x)›
5185      using "KBasic:11"[THEN "≡E"(1)] by blast
5186    AOT_hence ‹∀x ([F]x ≡ [λz [F]z & p1]x) & ◇¬∀x([F]x ≡ [λz [F]z & p1]x)›
5187      using Aux_A[THEN "→E", OF p1_prop[THEN "&E"(1)]] "&I" by blast
5188    AOT_hence ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5189      by (rule "∃I"(1)) "cqt:2[lambda]"
5190  }
5191  ultimately AOT_show ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5192    using "∨E"(1)[OF "exc-mid"] "→I" by blast
5193qed
5194
5195AOT_theorem "eqnotnec:5": ‹∀F∃G(¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5196proof(rule GEN)
5197  fix F
5198
5199  AOT_have Aux_A: ‹⊢□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5200  proof(rule "RM◇"; rule "→I"; rule GEN)
5201    AOT_modally_strict {
5202    fix x
5203    AOT_assume 0: ‹ψ›
5204    AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5205      by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5206    also AOT_have ‹... ≡ [F]x›
5207      apply (rule "≡I"; rule "→I")
5208      using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5209      using 0 "&I" by blast
5210    finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5211      using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5212    }
5213  qed
5214
5215  AOT_have Aux_B: ‹⊢□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5216  proof (rule "RM◇"; rule "→I"; rule GEN)
5217    AOT_modally_strict {
5218      fix x
5219      AOT_assume 0: ‹ψ›
5220      AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5221        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5222      also AOT_have ‹... ≡ [F]x›
5223        apply (rule "≡I"; rule "→I")
5224        using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5225        apply (rule "∨I"(1)) using 0 "&I" by blast
5226      finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5227        using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5228    }
5229  qed
5230
5231  AOT_have Aux_C: ‹⊢□ ¬ψ → ¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5232  proof(rule "→I"; rule "raa-cor:2")
5233  AOT_modally_strict {
5234      AOT_assume 0: ‹¬ψ›
5235      AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5236      AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5237      moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5238          by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5239      moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5240        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5241      ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5242        using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5243      moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5244      ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5245      AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5246    }
5247  qed
5248
5249  AOT_have Aux_D: ‹∀z ([F]z ≡ [λz [F]z & ψ]z) → (¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5250  proof (rule "→I"; rule "≡I"; (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5251    AOT_modally_strict {
5252      AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5253      AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5254      AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5255      AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5256      AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5257      AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5258    }
5259  next
5260    AOT_modally_strict {
5261      AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5262      AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5263      AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5264      AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5265      AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5266      AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5267    }
5268  qed
5269
5270  AOT_obtain p1 where p1_prop: ‹¬p1 & ◇p1› using "cont-tf-thm:2" "∃E"[rotated] "cont-tf:2"[THEN "≡dfE"] by blast
5271  {
5272    AOT_assume 1: ‹∀x([F]x ≡ [λz [F]z & p1]x)›
5273    AOT_have 2: ‹◇∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5274      using Aux_B[THEN "→E", OF p1_prop[THEN "&E"(2)]].
5275    AOT_have ‹¬∀x([λz [F]z & p1]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5276      using Aux_C[THEN "→E", OF p1_prop[THEN "&E"(1)]].
5277    AOT_hence 3: ‹¬∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5278      using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5279    AOT_hence ‹¬∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x) & ◇∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)› using 2 "&I" by blast
5280    AOT_hence ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5281      by (rule "∃I"(1)) "cqt:2[lambda]"
5282  }
5283  moreover {
5284    AOT_assume 2: ‹¬∀x([F]x ≡ [λz [F]z & p1]x)›
5285    AOT_hence ‹¬∀x([F]x ≡ [λz [F]z & p1]x)›
5286      using "KBasic:11"[THEN "≡E"(1)] by blast
5287    AOT_hence ‹¬∀x ([F]x ≡ [λz [F]z & p1]x) & ◇∀x([F]x ≡ [λz [F]z & p1]x)›
5288      using Aux_A[THEN "→E", OF p1_prop[THEN "&E"(2)]] "&I" by blast
5289    AOT_hence ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5290      by (rule "∃I"(1)) "cqt:2[lambda]"
5291  }
5292  ultimately AOT_show ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5293    using "∨E"(1)[OF "exc-mid"] "→I" by blast
5294qed
5295
5296AOT_theorem "eqnotnec:6": ‹∀F∃G(𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5297proof(rule GEN)
5298  fix F
5299
5300  AOT_have Aux_A: ‹⊢□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5301  proof(rule "RM◇"; rule "→I"; rule GEN)
5302    AOT_modally_strict {
5303    fix x
5304    AOT_assume 0: ‹ψ›
5305    AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5306      by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5307    also AOT_have ‹... ≡ [F]x›
5308      apply (rule "≡I"; rule "→I")
5309      using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5310      using 0 "&I" by blast
5311    finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5312      using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5313    }
5314  qed
5315
5316  AOT_have Aux_B: ‹⊢□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5317  proof (rule "RM◇"; rule "→I"; rule GEN)
5318    AOT_modally_strict {
5319      fix x
5320      AOT_assume 0: ‹ψ›
5321      AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5322        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5323      also AOT_have ‹... ≡ [F]x›
5324        apply (rule "≡I"; rule "→I")
5325        using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5326        apply (rule "∨I"(1)) using 0 "&I" by blast
5327      finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5328        using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5329    }
5330  qed
5331
5332  AOT_have Aux_C: ‹⊢□ 𝒜¬ψ → 𝒜¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5333  proof(rule "act-cond"[THEN "→E"]; rule "RA[2]"; rule "→I"; rule "raa-cor:2")
5334  AOT_modally_strict {
5335      AOT_assume 0: ‹¬ψ›
5336      AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5337      AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5338      moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5339          by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5340      moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5341        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5342      ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5343        using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5344      moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5345      ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5346      AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5347    }
5348  qed
5349
5350  AOT_have Aux_D: ‹𝒜∀z ([F]z ≡ [λz [F]z & ψ]z) → (𝒜¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ 𝒜¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5351  proof (rule "→I"; rule "Act-Basic:5"[THEN "≡E"(1)])
5352    AOT_assume ‹𝒜∀z ([F]z ≡ [λz [F]z & ψ]z)›
5353    AOT_thus ‹𝒜(¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))›
5354    proof (rule "RA[3]"[where Γ="{«∀z ([F]z ≡ [λz [F]z & ψ]z)»}", simplified, rotated])
5355      AOT_modally_strict {
5356        AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5357        AOT_thus ‹¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5358          apply -
5359        proof(rule "≡I"; (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5360        AOT_modally_strict {
5361          AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5362          AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5363          AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5364          AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5365          AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5366          AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5367        }
5368      next
5369        AOT_modally_strict {
5370          AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5371          AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5372          AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5373          AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5374          AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5375          AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5376        }
5377      qed
5378      }
5379    qed
5380  qed
5381
5382  AOT_have ‹¬𝒜q0›
5383    apply (rule "=dfI"(2)[OF q0_def])
5384     apply (fact "log-prop-prop:2")
5385    by (fact AOT)
5386  AOT_hence q0_prop_1: ‹𝒜¬q0›
5387    using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5388  {
5389    AOT_assume 1: ‹𝒜∀x([F]x ≡ [λz [F]z & q0]x)›
5390    AOT_have 2: ‹◇∀x([F]x ≡ [λz [F]z & q0 ∨ ¬q0]x)›
5391      using Aux_B[THEN "→E", OF q0_prop[THEN "&E"(1)]].
5392    AOT_have ‹𝒜¬∀x([λz [F]z & q0]x ≡ [λz [F]z & q0 ∨ ¬q0]x)›
5393      using Aux_C[THEN "→E", OF q0_prop_1].
5394    AOT_hence 3: ‹𝒜¬∀x([F]x ≡ [λz [F]z & q0 ∨ ¬q0]x)›
5395      using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5396    AOT_hence ‹𝒜¬∀x([F]x ≡ [λz [F]z & q0 ∨ ¬q0]x) & ◇∀x([F]x ≡ [λz [F]z & q0 ∨ ¬q0]x)› using 2 "&I" by blast
5397    AOT_hence ‹∃G (𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5398      by (rule "∃I"(1)) "cqt:2[lambda]"
5399  }
5400  moreover {
5401    AOT_assume 2: ‹¬𝒜∀x([F]x ≡ [λz [F]z & q0]x)›
5402    AOT_hence ‹𝒜¬∀x([F]x ≡ [λz [F]z & q0]x)›
5403      using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5404    AOT_hence ‹𝒜¬∀x ([F]x ≡ [λz [F]z & q0]x) & ◇∀x([F]x ≡ [λz [F]z & q0]x)›
5405      using Aux_A[THEN "→E", OF q0_prop[THEN "&E"(1)]] "&I" by blast
5406    AOT_hence ‹∃G (𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5407      by (rule "∃I"(1)) "cqt:2[lambda]"
5408  }
5409  ultimately AOT_show ‹∃G (𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5410    using "∨E"(1)[OF "exc-mid"] "→I" by blast
5411qed
5412
5413AOT_theorem "oa-contingent:1": ‹O! ≠ A!›
5414proof(rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
5415  fix x
5416  AOT_assume 1: ‹O! = A!›
5417  AOT_hence ‹[λx ◇E!x] = A!›
5418    by (rule "=dfE"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5419  AOT_hence ‹[λx ◇E!x] = [λx ¬◇E!x]›
5420    by (rule "=dfE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5421  moreover AOT_have ‹[λx ◇E!x]x ≡ ◇E!x›
5422    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5423  ultimately AOT_have ‹[λx ¬◇E!x]x ≡ ◇E!x›
5424    using "rule=E" by fast
5425  moreover AOT_have ‹[λx ¬◇E!x]x ≡ ¬◇E!x›
5426    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5427  ultimately AOT_have ‹◇E!x ≡ ¬◇E!x› using "≡E"(6) "Commutativity of ≡"[THEN "≡E"(1)] by blast
5428  AOT_thus "(◇E!x ≡ ¬◇E!x) & ¬(◇E!x ≡ ¬◇E!x)" using "oth-class-taut:3:c" "&I" by blast
5429qed
5430
5431AOT_theorem "oa-contingent:2": ‹O!x ≡ ¬A!x›
5432proof -
5433  AOT_have ‹O!x ≡ [λx ◇E!x]x›
5434    apply (rule "≡I"; rule "→I")
5435     apply (rule "=dfE"(2)[OF AOT_ordinary])
5436      apply "cqt:2[lambda]"
5437     apply argo
5438    apply (rule  "=dfI"(2)[OF AOT_ordinary])
5439     apply "cqt:2[lambda]"
5440    by argo
5441  also AOT_have ‹… ≡ ◇E!x›
5442    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5443  also AOT_have ‹… ≡ ¬¬◇E!x›
5444    using "oth-class-taut:3:b".
5445  also AOT_have ‹… ≡ ¬[λx ¬◇E!x]x›
5446    by (rule "beta-C-meta"[THEN "→E", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], symmetric]) "cqt:2[lambda]"
5447  also AOT_have ‹… ≡ ¬A!x›
5448    apply (rule "≡I"; rule "→I")
5449     apply (rule "=dfI"(2)[OF AOT_abstract])
5450      apply "cqt:2[lambda]"
5451     apply argo
5452    apply (rule "=dfE"(2)[OF AOT_abstract])
5453     apply "cqt:2[lambda]"
5454    by argo
5455  finally show ?thesis.
5456qed
5457
5458AOT_theorem "oa-contingent:3": ‹A!x ≡ ¬O!x›
5459  by (AOT_subst "«A!x»" "«¬¬A!x»")
5460     (auto simp add: "oth-class-taut:3:b" "oa-contingent:2"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], symmetric])
5461
5462AOT_theorem "oa-contingent:4": ‹Contingent(O!)›
5463proof (rule "thm-cont-prop:2"[unvarify F, OF "oa-exist:1", THEN "≡E"(2)]; rule "&I")
5464  AOT_have ‹◇∃x E!x› using "thm-cont-e:3" .
5465  AOT_hence ‹∃x ◇E!x› using "BF◇"[THEN "→E"] by blast
5466  then AOT_obtain a where ‹◇E!a› using "∃E"[rotated] by blast
5467  AOT_hence ‹[λx ◇E!x]a›
5468    by (rule "beta-C-meta"[THEN "→E", THEN "≡E"(2), rotated]) "cqt:2[lambda]"
5469  AOT_hence ‹O!a›
5470    by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5471  AOT_hence ‹∃x O!x› using "∃I" by blast
5472  AOT_thus ‹◇∃x O!x› using "T◇"[THEN "→E"] by blast
5473next
5474  AOT_obtain a where ‹A!a›
5475    using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5476  AOT_hence ‹¬O!a› using "oa-contingent:3"[THEN "≡E"(1)] by blast
5477  AOT_hence ‹∃x ¬O!x› using "∃I" by fast
5478  AOT_thus ‹◇∃x ¬O!x› using "T◇"[THEN "→E"] by blast
5479qed
5480
5481AOT_theorem "oa-contingent:5": ‹Contingent(A!)›
5482proof (rule "thm-cont-prop:2"[unvarify F, OF "oa-exist:2", THEN "≡E"(2)]; rule "&I")
5483  AOT_obtain a where ‹A!a›
5484    using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5485  AOT_hence ‹∃x A!x› using "∃I" by fast
5486  AOT_thus ‹◇∃x A!x› using "T◇"[THEN "→E"] by blast
5487next
5488  AOT_have ‹◇∃x E!x› using "thm-cont-e:3" .
5489  AOT_hence ‹∃x ◇E!x› using "BF◇"[THEN "→E"] by blast
5490  then AOT_obtain a where ‹◇E!a› using "∃E"[rotated] by blast
5491  AOT_hence ‹[λx ◇E!x]a›
5492    by (rule "beta-C-meta"[THEN "→E", THEN "≡E"(2), rotated]) "cqt:2[lambda]"
5493  AOT_hence ‹O!a›
5494    by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5495  AOT_hence ‹¬A!a› using "oa-contingent:2"[THEN "≡E"(1)] by blast
5496  AOT_hence ‹∃x ¬A!x› using "∃I" by fast
5497  AOT_thus ‹◇∃x ¬A!x› using "T◇"[THEN "→E"] by blast
5498qed
5499
5500AOT_theorem "oa-contingent:7": ‹O!-x ≡ ¬A!-x›
5501proof -
5502  AOT_have ‹O!x ≡ ¬A!x›
5503    using "oa-contingent:2" by blast
5504  also AOT_have ‹… ≡ A!-x›
5505    using "thm-relation-negation:1"[symmetric, unvarify F, OF "oa-exist:2"].
5506  finally AOT_have 1: ‹O!x ≡ A!-x›.
5507
5508  AOT_have ‹A!x ≡ ¬O!x›
5509    using "oa-contingent:3" by blast
5510  also AOT_have ‹… ≡ O!-x›
5511    using "thm-relation-negation:1"[symmetric, unvarify F, OF "oa-exist:1"].
5512  finally AOT_have 2: ‹A!x ≡ O!-x›.
5513
5514  AOT_show ‹O!-x ≡ ¬A!-x›
5515    using 1[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "oa-contingent:3"[of _ x] 2[symmetric]
5516          "≡E"(5) by blast
5517qed
5518
5519AOT_theorem "oa-contingent:6": ‹O!- ≠ A!-›
5520proof (rule "=-infix"[THEN "≡dfI"]; rule "raa-cor:2")
5521  AOT_assume 1: ‹O!- = A!-›
5522  fix x
5523  AOT_have ‹A!-x ≡ O!-x›
5524    apply (rule "rule=E"[rotated, OF 1]) by (fact "oth-class-taut:3:a")
5525  AOT_hence ‹A!-x ≡ ¬A!-x›
5526    using "oa-contingent:7" "≡E" by fast
5527  AOT_thus ‹(A!-x ≡ ¬A!-x) & ¬(A!-x ≡ ¬A!-x)› using "oth-class-taut:3:c" "&I" by blast
5528qed
5529
5530AOT_theorem "oa-contingent:8": ‹Contingent(O!-)›
5531  using "thm-cont-prop:3"[unvarify F, OF "oa-exist:1", THEN "≡E"(1), OF "oa-contingent:4"].
5532
5533AOT_theorem "oa-contingent:9": ‹Contingent(A!-)›
5534  using "thm-cont-prop:3"[unvarify F, OF "oa-exist:2", THEN "≡E"(1), OF "oa-contingent:5"].
5535
5536AOT_define WeaklyContingent :: ‹Π ⇒ φ› ("WeaklyContingent'(_')")
5537  "df-cont-nec": "WeaklyContingent([F]) ≡df Contingent([F]) & ∀x (◇[F]x → □[F]x)"
5538
5539AOT_theorem "cont-nec-fact1:1": ‹WeaklyContingent([F]) ≡ WeaklyContingent([F]-)›
5540proof -
5541  AOT_have ‹WeaklyContingent([F]) ≡ Contingent([F]) & ∀x (◇[F]x → □[F]x)›
5542    using "df-cont-nec"[THEN "≡Df"] by blast
5543  also AOT_have ‹... ≡ Contingent([F]-) & ∀x (◇[F]x → □[F]x)›
5544    apply (rule "oth-class-taut:8:f"[THEN "≡E"(2)]; rule "→I")
5545    using "thm-cont-prop:3".
5546  also AOT_have ‹… ≡ Contingent([F]-) & ∀x (◇[F]-x → □[F]-x)›
5547  proof (rule "oth-class-taut:8:e"[THEN "≡E"(2)]; rule "→I"; rule "≡I"; rule "→I"; rule GEN; rule "→I")
5548    fix x
5549    AOT_assume 0: ‹∀x (◇[F]x → □[F]x)›
5550    AOT_assume 1: ‹◇[F]-x›
5551    AOT_have ‹◇¬[F]x›
5552      by (AOT_subst_rev "«[F]-x»" "«¬[F]x»")
5553         (auto simp add: "thm-relation-negation:1" 1)
5554    AOT_hence 2: ‹¬□[F]x›
5555      using "KBasic:11"[THEN "≡E"(2)] by blast
5556    AOT_show ‹□[F]-x›
5557    proof (rule "raa-cor:1")
5558      AOT_assume 3: ‹¬□[F]-x›
5559      AOT_have ‹¬□¬[F]x›
5560        by (AOT_subst_rev "«[F]-x»" "«¬[F]x»")
5561           (auto simp add: "thm-relation-negation:1" 3)
5562      AOT_hence ‹◇[F]x›
5563        using "conventions:5"[THEN "≡dfI"] by simp
5564      AOT_hence ‹□[F]x› using 0 "∀E" "→E" by fast
5565      AOT_thus ‹□[F]x & ¬□[F]x› using "&I" 2 by blast
5566    qed
5567  next
5568    fix x
5569    AOT_assume 0: ‹∀x (◇[F]-x → □[F]-x)›
5570    AOT_assume 1: ‹◇[F]x›
5571    AOT_have ‹◇¬[F]-x›
5572      by (AOT_subst "«¬[F]-x»" "«[F]x»")
5573         (auto simp: "thm-relation-negation:2" 1)
5574    AOT_hence 2: ‹¬□[F]-x›
5575      using "KBasic:11"[THEN "≡E"(2)] by blast
5576    AOT_show ‹□[F]x›
5577    proof (rule "raa-cor:1")
5578      AOT_assume 3: ‹¬□[F]x›
5579      AOT_have ‹¬□¬[F]-x›
5580        by (AOT_subst "«¬[F]-x»" "«[F]x»")
5581           (auto simp add: "thm-relation-negation:2" 3)
5582      AOT_hence ‹◇[F]-x›
5583        using "conventions:5"[THEN "≡dfI"] by simp
5584      AOT_hence ‹□[F]-x› using 0 "∀E" "→E" by fast
5585      AOT_thus ‹□[F]-x & ¬□[F]-x› using "&I" 2 by blast
5586    qed
5587  qed
5588  also AOT_have ‹… ≡ WeaklyContingent([F]-)›
5589    using "df-cont-nec"[THEN "≡Df", symmetric] by blast
5590  finally show ?thesis.
5591qed
5592
5593AOT_theorem "cont-nec-fact1:2": ‹(WeaklyContingent([F]) & ¬WeaklyContingent([G])) → F ≠ G›
5594proof (rule "→I"; rule "=-infix"[THEN "≡dfI"]; rule "raa-cor:2")
5595  AOT_assume 1: ‹WeaklyContingent([F]) & ¬WeaklyContingent([G])›
5596  AOT_hence ‹WeaklyContingent([F])› using "&E" by blast
5597  moreover AOT_assume ‹F = G›
5598  ultimately AOT_have ‹WeaklyContingent([G])›
5599    using "rule=E" by blast
5600  AOT_thus ‹WeaklyContingent([G]) & ¬WeaklyContingent([G])›
5601    using 1 "&I" "&E" by blast
5602qed
5603
5604AOT_theorem "cont-nec-fact2:1": ‹WeaklyContingent(O!)›
5605proof (rule "df-cont-nec"[THEN "≡dfI"]; rule "&I")
5606  AOT_show ‹Contingent(O!)›
5607    using "oa-contingent:4".
5608next
5609  AOT_show ‹∀x (◇[O!]x → □[O!]x)›
5610    apply (rule GEN; rule "→I")
5611    using "oa-facts:5"[THEN "≡E"(1)] by blast
5612qed
5613
5614
5615AOT_theorem "cont-nec-fact2:2": ‹WeaklyContingent(A!)›
5616proof (rule "df-cont-nec"[THEN "≡dfI"]; rule "&I")
5617  AOT_show ‹Contingent(A!)›
5618    using "oa-contingent:5".
5619next
5620  AOT_show ‹∀x (◇[A!]x → □[A!]x)›
5621    apply (rule GEN; rule "→I")
5622    using "oa-facts:6"[THEN "≡E"(1)] by blast
5623qed
5624
5625AOT_theorem "cont-nec-fact2:3": ‹¬WeaklyContingent(E!)›
5626proof (rule "df-cont-nec"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)];
5627       rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(2); rule "raa-cor:2")
5628  AOT_have ‹◇∃x (E!x & ¬𝒜E!x)› using "qml:4"[axiom_inst].
5629  AOT_hence ‹∃x ◇(E!x & ¬𝒜E!x)› using "BF◇"[THEN "→E"] by blast
5630  then AOT_obtain a where ‹◇(E!a & ¬𝒜E!a)› using "∃E"[rotated] by blast
5631  AOT_hence 1: ‹◇E!a & ◇¬𝒜E!a› using "KBasic2:3"[THEN "→E"] by simp
5632  moreover AOT_assume ‹∀x (◇[E!]x → □[E!]x)›
5633  ultimately AOT_have ‹□E!a› using "&E" "∀E" "→E" by fast
5634  AOT_hence ‹𝒜E!a› using "nec-imp-act"[THEN "→E"] by blast
5635  AOT_hence ‹□𝒜E!a› using "qml-act:1"[axiom_inst, THEN "→E"] by blast
5636  moreover AOT_have ‹¬□𝒜E!a› using "KBasic:11"[THEN "≡E"(2)] 1[THEN "&E"(2)] by meson
5637  ultimately AOT_have ‹□𝒜E!a & ¬□𝒜E!a› using "&I" by blast
5638  AOT_thus ‹p & ¬p› for p using "raa-cor:1" by blast
5639qed
5640
5641AOT_theorem "cont-nec-fact2:4": ‹¬WeaklyContingent(L)›
5642  apply (rule "df-cont-nec"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)];
5643       rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(1))
5644  apply (rule "contingent-properties:4"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
5645  apply (rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(2); rule "useful-tautologies:2"[THEN "→E"])
5646  using "thm-noncont-e-e:3"[THEN "contingent-properties:3"[THEN "≡dfE"]].
5647
5648(* TODO: cleanup *)
5649AOT_theorem "cont-nec-fact2:5": ‹O! ≠ E! & O! ≠ E!- & O! ≠ L & O! ≠ L-›
5650proof -
5651  AOT_have 1: ‹L↓›
5652    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5653  {
5654    fix φ and Π and Π'
5655    AOT_have A: ‹¬(φ{Π'} ≡ φ{Π})› if  ‹φ{Π}› and ‹¬φ{Π'}›
5656    proof (rule "raa-cor:2")
5657      AOT_assume ‹φ{Π'} ≡ φ{Π}›
5658      AOT_hence ‹φ{Π'}› using that(1) "≡E" by blast
5659      AOT_thus ‹φ{Π'} & ¬φ{Π'}› using that(2) "&I" by blast
5660    qed
5661    AOT_have ‹Π' ≠ Π› if ‹Π↓› and ‹Π'↓› and ‹φ{Π}› and ‹¬φ{Π'}›
5662      using "pos-not-equiv-ne:4"[unvarify F G, THEN "→E", OF that(1,2), OF A[OF that(3, 4)]].
5663  } note 0 = this
5664  show ?thesis
5665    apply(safe intro!: "&I"; rule 0)
5666    using "cqt:2[concrete]"[axiom_inst] apply blast
5667    using "oa-exist:1" apply blast
5668    using "cont-nec-fact2:3" apply fast
5669    apply (rule "useful-tautologies:2"[THEN "→E"])
5670    using "cont-nec-fact2:1" apply fast
5671    using "rel-neg-T:3" apply fast
5672    using "oa-exist:1" apply blast
5673    using "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:3", OF "cqt:2[concrete]"[axiom_inst]] apply fast
5674    apply (rule "useful-tautologies:2"[THEN "→E"])
5675    using "cont-nec-fact2:1" apply blast
5676    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5677    using "oa-exist:1" apply fast
5678    using "cont-nec-fact2:4" apply fast
5679    apply (rule "useful-tautologies:2"[THEN "→E"])
5680    using "cont-nec-fact2:1" apply fast
5681    using "rel-neg-T:3" apply fast
5682    using "oa-exist:1" apply fast
5683    apply (rule "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:4"])
5684    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5685    apply (rule "useful-tautologies:2"[THEN "→E"])
5686    using "cont-nec-fact2:1" by blast
5687qed
5688
5689(* TODO: cleanup together with above *)
5690AOT_theorem "cont-nec-fact2:6": ‹A! ≠ E! & A! ≠ E!- & A! ≠ L & A! ≠ L-›
5691proof -
5692  AOT_have 1: ‹L↓›
5693    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5694  {
5695    fix φ and Π and Π'
5696    AOT_have A: ‹¬(φ{Π'} ≡ φ{Π})› if  ‹φ{Π}› and ‹¬φ{Π'}›
5697    proof (rule "raa-cor:2")
5698      AOT_assume ‹φ{Π'} ≡ φ{Π}›
5699      AOT_hence ‹φ{Π'}› using that(1) "≡E" by blast
5700      AOT_thus ‹φ{Π'} & ¬φ{Π'}› using that(2) "&I" by blast
5701    qed
5702    AOT_have ‹Π' ≠ Π› if ‹Π↓› and ‹Π'↓› and ‹φ{Π}› and ‹¬φ{Π'}›
5703      using "pos-not-equiv-ne:4"[unvarify F G, THEN "→E", OF that(1,2), OF A[OF that(3, 4)]].
5704  } note 0 = this
5705  show ?thesis
5706    apply(safe intro!: "&I"; rule 0)
5707    using "cqt:2[concrete]"[axiom_inst] apply blast
5708    using "oa-exist:2" apply blast
5709    using "cont-nec-fact2:3" apply fast
5710    apply (rule "useful-tautologies:2"[THEN "→E"])
5711    using "cont-nec-fact2:2" apply fast
5712    using "rel-neg-T:3" apply fast
5713    using "oa-exist:2" apply blast
5714    using "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:3", OF "cqt:2[concrete]"[axiom_inst]] apply fast
5715    apply (rule "useful-tautologies:2"[THEN "→E"])
5716    using "cont-nec-fact2:2" apply blast
5717    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5718    using "oa-exist:2" apply fast
5719    using "cont-nec-fact2:4" apply fast
5720    apply (rule "useful-tautologies:2"[THEN "→E"])
5721    using "cont-nec-fact2:2" apply fast
5722    using "rel-neg-T:3" apply fast
5723    using "oa-exist:2" apply fast
5724    apply (rule "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:4"])
5725    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5726    apply (rule "useful-tautologies:2"[THEN "→E"])
5727    using "cont-nec-fact2:2" by blast
5728qed
5729
5730AOT_define necessary_or_contingently_false :: ‹φ ⇒ φ› ("Δ_" [49] 54)
5731  ‹Δp ≡df □p ∨ (¬𝒜p & ◇p)›
5732
5733AOT_theorem sixteen:
5734 shows ‹∃F1∃F2∃F3∃F4∃F5∃F6∃F7∃F8∃F9∃F10∃F11∃F12∃F13∃F14∃F15∃F16 (
5735«F1::<κ>» ≠ F2 & F1 ≠ F3 & F1 ≠ F4 & F1 ≠ F5 & F1 ≠ F6 & F1 ≠ F7 & F1 ≠ F8 & F1 ≠ F9 & F1 ≠ F10 & F1 ≠ F11 & F1 ≠ F12 & F1 ≠ F13 & F1 ≠ F14 & F1 ≠ F15 & F1 ≠ F16 &
5736F2 ≠ F3 & F2 ≠ F4 & F2 ≠ F5 & F2 ≠ F6 & F2 ≠ F7 & F2 ≠ F8 & F2 ≠ F9 & F2 ≠ F10 & F2 ≠ F11 & F2 ≠ F12 & F2 ≠ F13 & F2 ≠ F14 & F2 ≠ F15 & F2 ≠ F16 &
5737F3 ≠ F4 & F3 ≠ F5 & F3 ≠ F6 & F3 ≠ F7 & F3 ≠ F8 & F3 ≠ F9 & F3 ≠ F10 & F3 ≠ F11 & F3 ≠ F12 & F3 ≠ F13 & F3 ≠ F14 & F3 ≠ F15 & F3 ≠ F16 &
5738F4 ≠ F5 & F4 ≠ F6 & F4 ≠ F7 & F4 ≠ F8 & F4 ≠ F9 & F4 ≠ F10 & F4 ≠ F11 & F4 ≠ F12 & F4 ≠ F13 & F4 ≠ F14 & F4 ≠ F15 & F4 ≠ F16 &
5739F5 ≠ F6 & F5 ≠ F7 & F5 ≠ F8 & F5 ≠ F9 & F5 ≠ F10 & F5 ≠ F11 & F5 ≠ F12 & F5 ≠ F13 & F5 ≠ F14 & F5 ≠ F15 & F5 ≠ F16 &
5740F6 ≠ F7 & F6 ≠ F8 & F6 ≠ F9 & F6 ≠ F10 & F6 ≠ F11 & F6 ≠ F12 & F6 ≠ F13 & F6 ≠ F14 & F6 ≠ F15 & F6 ≠ F16 &
5741F7 ≠ F8 & F7 ≠ F9 & F7 ≠ F10 & F7 ≠ F11 & F7 ≠ F12 & F7 ≠ F13 & F7 ≠ F14 & F7 ≠ F15 & F7 ≠ F16 &
5742F8 ≠ F9 & F8 ≠ F10 & F8 ≠ F11 & F8 ≠ F12 & F8 ≠ F13 & F8 ≠ F14 & F8 ≠ F15 & F8 ≠ F16 &
5743F9 ≠ F10 & F9 ≠ F11 & F9 ≠ F12 & F9 ≠ F13 & F9 ≠ F14 & F9 ≠ F15 & F9 ≠ F16 &
5744F10 ≠ F11 & F10 ≠ F12 & F10 ≠ F13 & F10 ≠ F14 & F10 ≠ F15 & F10 ≠ F16 &
5745F11 ≠ F12 & F11 ≠ F13 & F11 ≠ F14 & F11 ≠ F15 & F11 ≠ F16 &
5746F12 ≠ F13 & F12 ≠ F14 & F12 ≠ F15 & F12 ≠ F16 &
5747F13 ≠ F14 & F13 ≠ F15 & F13 ≠ F16 &
5748F14 ≠ F15 & F14 ≠ F16 &
5749F15 ≠ F16)› 
5750proof -
5751
5752  AOT_have Delta_pos: ‹Δφ → ◇φ› for φ
5753  proof(rule "→I")
5754    AOT_assume ‹Δφ›
5755    AOT_hence ‹□φ ∨ (¬𝒜φ & ◇φ)›
5756      using "≡dfE"[OF necessary_or_contingently_false] by blast
5757    moreover {
5758      AOT_assume ‹□φ›
5759      AOT_hence ‹◇φ›
5760        by (metis "B◇" "T◇" "vdash-properties:10")
5761    }
5762    moreover {
5763      AOT_assume ‹¬𝒜φ & ◇φ›
5764      AOT_hence ‹◇φ›
5765        using "&E" by blast
5766    }
5767    ultimately AOT_show ‹◇φ›
5768      by (metis "∨E"(2) "raa-cor:1") 
5769  qed
5770
5771  AOT_have act_and_not_nec_not_delta: ‹¬Δφ› if ‹𝒜φ› and ‹¬□φ› for φ
5772    using "≡dfE" "&E"(1) "∨E"(2) necessary_or_contingently_false "raa-cor:3" that(1) that(2) by blast
5773  AOT_have act_and_pos_not_not_delta: ‹¬Δφ› if ‹𝒜φ› and ‹◇¬φ› for φ
5774    using "KBasic:11" act_and_not_nec_not_delta "≡E"(2) that(1) that(2) by blast
5775  AOT_have impossible_delta: ‹¬Δφ› if ‹¬◇φ› for φ
5776    using Delta_pos "modus-tollens:1" that by blast
5777  AOT_have not_act_and_pos_delta: ‹Δφ› if ‹¬𝒜φ› and ‹◇φ› for φ
5778    by (meson "≡dfI" "&I" "∨I"(2) necessary_or_contingently_false that(1) that(2))
5779  AOT_have nec_delta: ‹Δφ› if ‹□φ› for φ
5780    using "≡dfI" "∨I"(1) necessary_or_contingently_false that by blast
5781
5782  AOT_obtain a where a_prop: ‹A!a›
5783    using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5784  AOT_obtain b where b_prop: ‹◇[E!]b & ¬𝒜[E!]b›
5785    using "pos-not-pna:3" using "∃E"[rotated] by blast
5786
5787  AOT_have b_ord: ‹[O!]b›
5788  proof(rule "=dfI"(2)[OF AOT_ordinary])
5789    AOT_show ‹[λx ◇[E!]x]↓› by "cqt:2[lambda]"
5790  next
5791    AOT_show ‹[λx ◇[E!]x]b›
5792    proof (rule "β←C"(1); ("cqt:2[lambda]")?)
5793      AOT_show ‹b↓› by (rule "cqt:2[const_var]"[axiom_inst])
5794      AOT_show ‹◇[E!]b› by (fact b_prop[THEN "&E"(1)])
5795    qed
5796  qed
5797
5798  AOT_have nec_not_L_neg: ‹□¬[L-]x› for x
5799    using "thm-noncont-e-e:2" "contingent-properties:2"[THEN "≡dfE"] "&E"
5800          CBF[THEN "→E"] "∀E" by blast
5801  AOT_have nec_L: ‹□[L]x› for x
5802    using "thm-noncont-e-e:1" "contingent-properties:1"[THEN "≡dfE"]
5803      CBF[THEN "→E"] "∀E" by blast
5804
5805  AOT_have act_ord_b: ‹𝒜[O!]b›
5806    using b_ord "≡E"(1) "oa-facts:7" by blast
5807  AOT_have delta_ord_b: ‹Δ[O!]b›
5808    by (meson "≡dfI" b_ord "∨I"(1) necessary_or_contingently_false "oa-facts:1" "vdash-properties:10")
5809  AOT_have not_act_ord_a: ‹¬𝒜[O!]a›
5810    by (meson a_prop "≡E"(1) "≡E"(3) "oa-contingent:3" "oa-facts:7")
5811  AOT_have not_delta_ord_a: ‹¬Δ[O!]a›
5812    by (metis Delta_pos "≡E"(4) not_act_ord_a "oa-facts:3" "oa-facts:7" "reductio-aa:1" "vdash-properties:10")
5813
5814  AOT_have not_act_abs_b: ‹¬𝒜[A!]b›
5815    by (meson b_ord "≡E"(1) "≡E"(3) "oa-contingent:2" "oa-facts:8")
5816  AOT_have not_delta_abs_b: ‹¬Δ[A!]b›
5817  proof(rule "raa-cor:2")
5818    AOT_assume ‹Δ[A!]b›
5819    AOT_hence ‹◇[A!]b›
5820      by (metis Delta_pos "vdash-properties:10")
5821    AOT_thus ‹[A!]b & ¬[A!]b›
5822      by (metis b_ord "&I" "≡E"(1) "oa-contingent:2" "oa-facts:4" "vdash-properties:10")
5823  qed
5824  AOT_have act_abs_a: ‹𝒜[A!]a›
5825    using a_prop "≡E"(1) "oa-facts:8" by blast
5826  AOT_have delta_abs_a: ‹Δ[A!]a›
5827      by (metis "≡dfI" a_prop "oa-facts:2" "vdash-properties:10" "∨I"(1) necessary_or_contingently_false)
5828
5829  AOT_have not_act_concrete_b: ‹¬𝒜[E!]b›
5830    using b_prop "&E"(2) by blast
5831  AOT_have delta_concrete_b: ‹Δ[E!]b›
5832  proof (rule "≡dfI"[OF necessary_or_contingently_false]; rule "∨I"(2); rule "&I")
5833    AOT_show ‹¬𝒜[E!]b› using b_prop "&E"(2) by blast
5834  next
5835    AOT_show ‹◇[E!]b› using b_prop "&E"(1) by blast
5836  qed
5837  AOT_have not_act_concrete_a: ‹¬𝒜[E!]a›
5838  proof (rule "raa-cor:2")
5839    AOT_assume ‹𝒜[E!]a›
5840    AOT_hence 1: ‹◇[E!]a› by (metis "Act-Sub:3" "vdash-properties:10")
5841    AOT_have ‹[A!]a› by (simp add: a_prop)
5842    AOT_hence ‹[λx ¬◇[E!]x]a›
5843      by (rule "=dfE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5844    AOT_hence ‹¬◇[E!]a› using "β→C"(1) by blast
5845    AOT_thus ‹◇[E!]a & ¬◇[E!]a› using 1 "&I" by blast
5846  qed
5847  AOT_have not_delta_concrete_a: ‹¬Δ[E!]a›
5848  proof (rule "raa-cor:2")
5849    AOT_assume ‹Δ[E!]a›
5850    AOT_hence 1: ‹◇[E!]a› by (metis Delta_pos "vdash-properties:10")
5851    AOT_have ‹[A!]a› by (simp add: a_prop)
5852    AOT_hence ‹[λx ¬◇[E!]x]a›
5853      by (rule "=dfE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5854    AOT_hence ‹¬◇[E!]a› using "β→C"(1) by blast
5855    AOT_thus ‹◇[E!]a & ¬◇[E!]a› using 1 "&I" by blast
5856  qed
5857
5858  AOT_have not_act_q_zero: ‹¬𝒜q0›
5859    by (meson "log-prop-prop:2" "pos-not-pna:1" q0_def "reductio-aa:1" "rule-id-def:2:a[zero]")
5860  AOT_have delta_q_zero: ‹Δq0›
5861  proof(rule "≡dfI"[OF necessary_or_contingently_false]; rule "∨I"(2); rule "&I")
5862    AOT_show ‹¬𝒜q0› using not_act_q_zero.
5863    AOT_show ‹◇q0› by (meson "&E"(1) q0_prop)
5864  qed
5865  AOT_have act_not_q_zero: ‹𝒜¬q0› using "Act-Basic:1" "∨E"(2) not_act_q_zero by blast
5866  AOT_have not_delta_not_q_zero: ‹¬Δ¬q0›
5867      using "≡dfE" "conventions:5" "Act-Basic:1" act_and_not_nec_not_delta "&E"(1) "∨E"(2) not_act_q_zero q0_prop by blast
5868
5869  AOT_have ‹[L-]↓› by (simp add: "rel-neg-T:3")
5870  moreover AOT_have ‹¬𝒜[L-]b & ¬Δ[L-]b & ¬𝒜[L-]a & ¬Δ[L-]a›
5871  proof (safe intro!: "&I")
5872    AOT_show ‹¬𝒜[L-]b› by (meson "≡E"(1) "logic-actual-nec:1"[axiom_inst] "nec-imp-act" nec_not_L_neg "→E")
5873    AOT_show ‹¬Δ[L-]b› by (meson Delta_pos "KBasic2:1" "≡E"(1) "modus-tollens:1" nec_not_L_neg)
5874    AOT_show ‹¬𝒜[L-]a› by (meson "≡E"(1) "logic-actual-nec:1"[axiom_inst] "nec-imp-act" nec_not_L_neg "→E")
5875    AOT_show ‹¬Δ[L-]a› using Delta_pos "KBasic2:1" "≡E"(1) "modus-tollens:1" nec_not_L_neg by blast
5876  qed
5877  ultimately AOT_obtain F0 where ‹¬𝒜[F0]b & ¬Δ[F0]b & ¬𝒜[F0]a & ¬Δ[F0]a›
5878    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5879  AOT_hence ‹¬𝒜[F0]b› and ‹¬Δ[F0]b› and ‹¬𝒜[F0]a› and ‹¬Δ[F0]a›
5880    using "&E" by blast+
5881  note props = this
5882
5883  let ?Π = "«[λy [A!]y & q0]»"
5884  AOT_modally_strict {
5885    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
5886  } note 1 = this
5887  moreover AOT_have‹¬𝒜[«?Π»]b & ¬Δ[«?Π»]b & ¬𝒜[«?Π»]a & Δ[«?Π»]a›
5888  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
5889    AOT_show ‹¬𝒜([A!]b & q0)›
5890      using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_abs_b "raa-cor:3" by blast
5891  next AOT_show ‹¬Δ([A!]b & q0)›
5892      by (metis Delta_pos "KBasic2:3" "&E"(1) "≡E"(4) not_act_abs_b "oa-facts:4" "oa-facts:8" "raa-cor:3" "vdash-properties:10")
5893  next AOT_show ‹¬𝒜([A!]a & q0)›
5894      using "Act-Basic:2" "&E"(2) "≡E"(1) not_act_q_zero "raa-cor:3" by blast
5895  next AOT_show ‹Δ([A!]a & q0)›
5896    proof (rule not_act_and_pos_delta)
5897      AOT_show ‹¬𝒜([A!]a & q0)›
5898        using "Act-Basic:2" "&E"(2) "≡E"(4) not_act_q_zero "raa-cor:3" by blast
5899    next AOT_show ‹◇([A!]a & q0)›
5900        by (metis "&I" "→E" Delta_pos "KBasic:16" "&E"(1) delta_abs_a "≡E"(1) "oa-facts:6" q0_prop)
5901    qed
5902  qed
5903  ultimately AOT_obtain F1 where ‹¬𝒜[F1]b & ¬Δ[F1]b & ¬𝒜[F1]a & Δ[F1]a›
5904    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5905  AOT_hence ‹¬𝒜[F1]b› and ‹¬Δ[F1]b› and ‹¬𝒜[F1]a› and ‹Δ[F1]a›
5906    using "&E" by blast+
5907  note props = props this
5908
5909  let ?Π = "«[λy [A!]y & ¬q0]»"
5910  AOT_modally_strict {
5911    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
5912  } note 1 = this
5913  moreover AOT_have ‹¬𝒜[«?Π»]b & ¬Δ[«?Π»]b & 𝒜[«?Π»]a & ¬Δ[«?Π»]a›
5914  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
5915    AOT_show ‹¬𝒜([A!]b & ¬q0)›
5916      using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_abs_b "raa-cor:3" by blast
5917  next AOT_show ‹¬Δ([A!]b & ¬q0)›
5918      by (meson "RM◇" Delta_pos "Conjunction Simplification"(1) "≡E"(4) "modus-tollens:1" not_act_abs_b "oa-facts:4" "oa-facts:8")
5919  next AOT_show ‹𝒜([A!]a & ¬q0)›
5920      by (metis "Act-Basic:1" "Act-Basic:2" act_abs_a "&I" "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:3")
5921  next AOT_show ‹¬Δ([A!]a & ¬q0)›
5922    proof (rule act_and_not_nec_not_delta)
5923      AOT_show ‹𝒜([A!]a & ¬q0)›
5924        by (metis "Act-Basic:1" "Act-Basic:2" act_abs_a "&I" "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:3")
5925    next
5926      AOT_show ‹¬□([A!]a & ¬q0)›
5927        by (metis "KBasic2:1" "KBasic:3" "&E"(1) "&E"(2) "≡E"(4) q0_prop "raa-cor:3")
5928    qed
5929  qed
5930  ultimately AOT_obtain F2 where ‹¬𝒜[F2]b & ¬Δ[F2]b & 𝒜[F2]a & ¬Δ[F2]a›
5931    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5932  AOT_hence ‹¬𝒜[F2]b› and ‹¬Δ[F2]b› and ‹𝒜[F2]a› and ‹¬Δ[F2]a›
5933    using "&E" by blast+
5934  note props = props this
5935
5936  AOT_have abstract_prop: ‹¬𝒜[A!]b & ¬Δ[A!]b & 𝒜[A!]a & Δ[A!]a›
5937    using act_abs_a "&I" delta_abs_a not_act_abs_b not_delta_abs_b by presburger
5938  then AOT_obtain F3 where ‹¬𝒜[F3]b & ¬Δ[F3]b & 𝒜[F3]a & Δ[F3]a›
5939    using "∃I"(1)[rotated, THEN "∃E"[rotated]] "oa-exist:2" by fastforce
5940  AOT_hence ‹¬𝒜[F3]b› and ‹¬Δ[F3]b› and ‹𝒜[F3]a› and ‹Δ[F3]a›
5941    using "&E" by blast+
5942  note props = props this
5943
5944  AOT_have ‹¬𝒜[E!]b & Δ[E!]b & ¬𝒜[E!]a & ¬Δ[E!]a›
5945    by (meson "&I" delta_concrete_b not_act_concrete_a not_act_concrete_b not_delta_concrete_a)
5946  then AOT_obtain F4 where ‹¬𝒜[F4]b & Δ[F4]b & ¬𝒜[F4]a & ¬Δ[F4]a›
5947    using "cqt:2[concrete]"[axiom_inst] "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5948  AOT_hence ‹¬𝒜[F4]b› and ‹Δ[F4]b› and ‹¬𝒜[F4]a› and ‹¬Δ[F4]a›
5949    using "&E" by blast+
5950  note props = props this
5951
5952  AOT_modally_strict {
5953    AOT_have ‹[λy q0]↓› by "cqt:2[lambda]"
5954  } note 1 = this
5955  moreover AOT_have ‹¬𝒜[λy q0]b & Δ[λy q0]b & ¬𝒜[λy q0]a & Δ[λy q0]a›
5956    by (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
5957       (auto simp: not_act_q_zero delta_q_zero)
5958  ultimately AOT_obtain F5 where ‹¬𝒜[F5]b & Δ[F5]b & ¬𝒜[F5]a & Δ[F5]a›
5959    using "cqt:2[concrete]"[axiom_inst] "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5960  AOT_hence ‹¬𝒜[F5]b› and ‹Δ[F5]b› and ‹¬𝒜[F5]a› and ‹Δ[F5]a›
5961    using "&E" by blast+
5962  note props = props this
5963
5964  let ?Π = "«[λy [E!]y ∨ ([A!]y & ¬q0)]»"
5965  AOT_modally_strict {
5966    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
5967  } note 1 = this
5968  moreover AOT_have ‹¬𝒜[«?Π»]b & Δ[«?Π»]b & 𝒜[«?Π»]a & ¬Δ[«?Π»]a›
5969  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
5970    AOT_have ‹𝒜¬([A!]b & ¬q0)›
5971      by (metis "Act-Basic:1" "Act-Basic:2" abstract_prop "&E"(1) "∨E"(2)
5972                "≡E"(1) "raa-cor:3")
5973    moreover AOT_have ‹¬𝒜[E!]b›
5974      using b_prop "&E"(2) by blast
5975    ultimately AOT_have 2: ‹𝒜(¬[E!]b & ¬([A!]b & ¬q0))›
5976      by (metis "Act-Basic:2" "Act-Sub:1" "&I" "≡E"(3) "raa-cor:1")
5977    AOT_have ‹𝒜¬([E!]b ∨ ([A!]b & ¬q0))›
5978      by (AOT_subst ‹«¬([E!]b ∨ ([A!]b & ¬q0))»› ‹«¬[E!]b & ¬([A!]b & ¬q0)»›)
5979         (auto simp: "oth-class-taut:5:d" 2)
5980    AOT_thus ‹¬𝒜([E!]b ∨ ([A!]b & ¬q0))›
5981      by (metis "¬¬I" "Act-Sub:1" "≡E"(4))
5982  next
5983    AOT_show ‹Δ([E!]b ∨ ([A!]b & ¬q0))›
5984    proof (rule not_act_and_pos_delta)
5985      AOT_show ‹¬𝒜([E!]b ∨ ([A!]b & ¬q0))›
5986        by (metis "Act-Basic:2" "Act-Basic:9" "∨E"(2) "Conjunction Simplification"(1) "≡E"(4) "modus-tollens:1" not_act_abs_b not_act_concrete_b "raa-cor:3")
5987    next
5988      AOT_show ‹◇([E!]b ∨ ([A!]b & ¬q0))›
5989        using "KBasic2:2" b_prop "&E"(1) "∨I"(1) "≡E"(3) "raa-cor:3" by blast
5990    qed
5991  next AOT_show ‹𝒜([E!]a ∨ ([A!]a & ¬q0))›
5992      by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_abs_a "&I" "∨I"(2) "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:1")
5993  next AOT_show ‹¬Δ([E!]a ∨ ([A!]a & ¬q0))›
5994    proof (rule act_and_not_nec_not_delta)
5995      AOT_show ‹𝒜([E!]a ∨ ([A!]a & ¬q0))›
5996        by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_abs_a "&I" "∨I"(2) "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:1")
5997    next
5998      AOT_have ‹□¬[E!]a›
5999        by (metis "≡dfI" "conventions:5" "&I" "∨I"(2) necessary_or_contingently_false not_act_concrete_a not_delta_concrete_a "raa-cor:3")
6000      moreover AOT_have ‹◇¬([A!]a & ¬q0)›
6001        by (metis "KBasic2:1" "KBasic:11" "KBasic:3" "&E"(1) "&E"(2) "≡E"(1) q0_prop "raa-cor:3")
6002      ultimately AOT_have ‹◇(¬[E!]a & ¬([A!]a & ¬q0))› by (metis "KBasic:16" "&I" "vdash-properties:10")
6003      AOT_hence ‹◇¬([E!]a ∨ ([A!]a & ¬q0))›
6004        by (metis "RE◇" "≡E"(2) "oth-class-taut:5:d")
6005      AOT_thus ‹¬□([E!]a ∨ ([A!]a & ¬q0))› by (metis "KBasic:12" "≡E"(1) "raa-cor:3")
6006    qed
6007  qed
6008  ultimately AOT_obtain F6 where ‹¬𝒜[F6]b & Δ[F6]b & 𝒜[F6]a & ¬Δ[F6]a›
6009    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6010  AOT_hence ‹¬𝒜[F6]b› and ‹Δ[F6]b› and ‹𝒜[F6]a› and ‹¬Δ[F6]a›
6011    using "&E" by blast+
6012  note props = props this
6013
6014  let ?Π = "«[λy [A!]y ∨ [E!]y]»"
6015  AOT_modally_strict {
6016    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6017  } note 1 = this
6018  moreover AOT_have ‹¬𝒜[«?Π»]b & Δ[«?Π»]b & 𝒜[«?Π»]a & Δ[«?Π»]a›
6019  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6020    AOT_show ‹¬𝒜([A!]b ∨ [E!]b)›
6021      using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_abs_b not_act_concrete_b "raa-cor:3" by blast
6022  next AOT_show ‹Δ([A!]b ∨ [E!]b)›
6023    proof (rule not_act_and_pos_delta)
6024      AOT_show ‹¬𝒜([A!]b ∨ [E!]b)›
6025        using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_abs_b not_act_concrete_b "raa-cor:3" by blast
6026    next AOT_show ‹◇([A!]b ∨ [E!]b)›
6027        using "KBasic2:2" b_prop "&E"(1) "∨I"(2) "≡E"(2) by blast
6028    qed
6029  next AOT_show ‹𝒜([A!]a ∨ [E!]a)›
6030      by (meson "Act-Basic:9" act_abs_a "∨I"(1) "≡E"(2))
6031  next AOT_show ‹Δ([A!]a ∨ [E!]a) ›
6032    proof (rule nec_delta)
6033      AOT_show ‹□([A!]a ∨ [E!]a)›
6034        by (metis "KBasic:15" act_abs_a act_and_not_nec_not_delta "Disjunction Addition"(1) delta_abs_a "raa-cor:3" "vdash-properties:10")
6035    qed
6036  qed
6037  ultimately AOT_obtain F7 where ‹¬𝒜[F7]b & Δ[F7]b & 𝒜[F7]a & Δ[F7]a›
6038    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6039  AOT_hence ‹¬𝒜[F7]b› and ‹Δ[F7]b› and ‹𝒜[F7]a› and ‹Δ[F7]a›
6040    using "&E" by blast+
6041  note props = props this
6042
6043  let ?Π = "«[λy [O!]y & ¬[E!]y]»"
6044  AOT_modally_strict {
6045    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6046  } note 1 = this
6047  moreover AOT_have ‹𝒜[«?Π»]b & ¬Δ[«?Π»]b & ¬𝒜[«?Π»]a & ¬Δ[«?Π»]a›
6048  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6049    AOT_show ‹𝒜([O!]b & ¬[E!]b)›
6050      by (metis "Act-Basic:1" "Act-Basic:2" act_ord_b "&I" "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:3")
6051  next AOT_show ‹¬Δ([O!]b & ¬[E!]b)›
6052      by (metis (no_types, hide_lams) "conventions:5" "Act-Sub:1" "RM:1" act_and_not_nec_not_delta "act-conj-act:3"
6053                act_ord_b b_prop "&I" "&E"(1) "Conjunction Simplification"(2) "df-rules-formulas[3]"
6054                "≡E"(3) "raa-cor:1" "→E")
6055  next AOT_show ‹¬𝒜([O!]a & ¬[E!]a)›
6056      using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_ord_a "raa-cor:3" by blast
6057  next AOT_have ‹¬◇([O!]a & ¬[E!]a)›
6058      by (metis "KBasic2:3" "&E"(1) "≡E"(4) not_act_ord_a "oa-facts:3" "oa-facts:7" "raa-cor:3" "vdash-properties:10")
6059    AOT_thus ‹¬Δ([O!]a & ¬[E!]a)›
6060      by (rule impossible_delta)
6061  qed      
6062  ultimately AOT_obtain F8 where ‹𝒜[F8]b & ¬Δ[F8]b & ¬𝒜[F8]a & ¬Δ[F8]a›
6063    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6064  AOT_hence ‹𝒜[F8]b› and ‹¬Δ[F8]b› and ‹¬𝒜[F8]a› and ‹¬Δ[F8]a›
6065    using "&E" by blast+
6066  note props = props this
6067
6068  (* TODO_PLM: binary property 9 wrong in PLM *)
6069  let ?Π = "«[λy ¬[E!]y & ([O!]y ∨ q0)]»"
6070  AOT_modally_strict {
6071    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6072  } note 1 = this
6073  moreover AOT_have ‹𝒜[«?Π»]b & ¬Δ[«?Π»]b & ¬𝒜[«?Π»]a & Δ[«?Π»]a›
6074  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6075    AOT_show ‹𝒜(¬[E!]b & ([O!]b ∨ q0))›
6076      by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_ord_b "&I" "∨I"(1)
6077                "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:1")
6078  next AOT_show ‹¬Δ(¬[E!]b & ([O!]b ∨ q0))›
6079    proof (rule act_and_pos_not_not_delta)
6080      AOT_show ‹𝒜(¬[E!]b & ([O!]b ∨ q0))›
6081        by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_ord_b "&I" "∨I"(1)
6082                  "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:1")
6083    next
6084      AOT_show ‹◇¬(¬[E!]b & ([O!]b ∨ q0))›
6085      proof (AOT_subst ‹«¬(¬[E!]b & ([O!]b ∨ q0))»› ‹«[E!]b ∨ ¬([O!]b ∨ q0)»›)
6086        AOT_modally_strict {
6087          AOT_show ‹¬(¬[E!]b & ([O!]b ∨ q0)) ≡ [E!]b ∨ ¬([O!]b ∨ q0)›
6088            by (metis "&I" "&E"(1) "&E"(2) "∨I"(1) "∨I"(2) "∨E"(2) "deduction-theorem" "≡I" "reductio-aa:1")
6089        }
6090      next
6091        AOT_show ‹◇([E!]b ∨ ¬([O!]b ∨ q0))›
6092          using "KBasic2:2" b_prop "&E"(1) "∨I"(1) "≡E"(3) "raa-cor:3" by blast
6093       qed
6094     qed
6095   next
6096     AOT_show ‹¬𝒜(¬[E!]a & ([O!]a ∨ q0))›
6097       using "Act-Basic:2" "Act-Basic:9" "&E"(2) "∨E"(3) "≡E"(1) not_act_ord_a not_act_q_zero "reductio-aa:2" by blast
6098   next
6099     AOT_show ‹Δ(¬[E!]a & ([O!]a ∨ q0))›
6100     proof (rule not_act_and_pos_delta)
6101       AOT_show ‹¬𝒜(¬[E!]a & ([O!]a ∨ q0))›
6102         by (metis "Act-Basic:2" "Act-Basic:9" "&E"(2) "∨E"(3) "≡E"(1) not_act_ord_a not_act_q_zero "reductio-aa:2")
6103     next
6104       AOT_have ‹□¬[E!]a›
6105         using "KBasic2:1" "≡E"(2) not_act_and_pos_delta not_act_concrete_a not_delta_concrete_a "raa-cor:5" by blast
6106       moreover AOT_have ‹◇([O!]a ∨ q0)›
6107         by (metis "KBasic2:2" "&E"(1) "∨I"(2) "≡E"(3) q0_prop "raa-cor:3")
6108       ultimately AOT_show ‹◇(¬[E!]a & ([O!]a ∨ q0))›
6109         by (metis "KBasic:16" "&I" "vdash-properties:10")
6110     qed
6111   qed
6112  ultimately AOT_obtain F9 where ‹𝒜[F9]b & ¬Δ[F9]b & ¬𝒜[F9]a & Δ[F9]a›
6113    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6114  AOT_hence ‹𝒜[F9]b› and ‹¬Δ[F9]b› and ‹¬𝒜[F9]a› and ‹Δ[F9]a›
6115    using "&E" by blast+
6116  note props = props this
6117
6118  AOT_modally_strict {
6119    AOT_have ‹[λy ¬q0]↓› by "cqt:2[lambda]"
6120  } note 1 = this
6121  moreover AOT_have ‹𝒜[λy ¬q0]b & ¬Δ[λy ¬q0]b & 𝒜[λy ¬q0]a & ¬Δ[λy ¬q0]a›
6122    by (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1]; auto simp: act_not_q_zero not_delta_not_q_zero)
6123  ultimately AOT_obtain F10 where ‹𝒜[F10]b & ¬Δ[F10]b & 𝒜[F10]a & ¬Δ[F10]a›
6124    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6125  AOT_hence ‹𝒜[F10]b› and ‹¬Δ[F10]b› and ‹𝒜[F10]a› and ‹¬Δ[F10]a›
6126    using "&E" by blast+
6127  note props = props this
6128
6129  AOT_modally_strict {
6130    AOT_have ‹[λy ¬[E!]y]↓› by "cqt:2[lambda]"
6131  } note 1 = this
6132  moreover AOT_have ‹𝒜[λy ¬[E!]y]b & ¬Δ[λy ¬[E!]y]b & 𝒜[λy ¬[E!]y]a & Δ[λy ¬[E!]y]a›
6133  proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6134    AOT_show ‹𝒜¬[E!]b›
6135      using "Act-Basic:1" "∨E"(2) not_act_concrete_b by blast
6136  next AOT_show ‹¬Δ¬[E!]b›
6137      using "≡dfE" "conventions:5" "Act-Basic:1" act_and_not_nec_not_delta b_prop "&E"(1) "∨E"(2) not_act_concrete_b by blast
6138  next AOT_show ‹𝒜¬[E!]a›
6139      using "Act-Basic:1" "∨E"(2) not_act_concrete_a by blast
6140  next AOT_show ‹Δ¬[E!]a›
6141      using "KBasic2:1" "≡E"(2) nec_delta not_act_and_pos_delta not_act_concrete_a not_delta_concrete_a "reductio-aa:1" by blast
6142  qed
6143  ultimately AOT_obtain F11 where ‹𝒜[F11]b & ¬Δ[F11]b & 𝒜[F11]a & Δ[F11]a›
6144    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6145  AOT_hence ‹𝒜[F11]b› and ‹¬Δ[F11]b› and ‹𝒜[F11]a› and ‹Δ[F11]a›
6146    using "&E" by blast+
6147  note props = props this
6148
6149  AOT_have ‹𝒜[O!]b & Δ[O!]b & ¬𝒜[O!]a & ¬Δ[O!]a›
6150    by (simp add: act_ord_b "&I" delta_ord_b not_act_ord_a not_delta_ord_a)
6151  then AOT_obtain F12 where ‹𝒜[F12]b & Δ[F12]b & ¬𝒜[F12]a & ¬Δ[F12]a›
6152    using "oa-exist:1" "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6153  AOT_hence ‹𝒜[F12]b› and ‹Δ[F12]b› and ‹¬𝒜[F12]a› and ‹¬Δ[F12]a›
6154    using "&E" by blast+
6155  note props = props this
6156
6157  let ?Π = "«[λy [O!]y ∨ q0]»"
6158  AOT_modally_strict {
6159    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6160  } note 1 = this
6161  moreover AOT_have ‹𝒜[«?Π»]b & Δ[«?Π»]b & ¬𝒜[«?Π»]a & Δ[«?Π»]a›
6162  proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6163    AOT_show ‹𝒜([O!]b ∨ q0)›
6164      by (meson "Act-Basic:9" act_ord_b "∨I"(1) "≡E"(2))
6165  next AOT_show ‹Δ([O!]b ∨ q0)›
6166      by (meson "KBasic:15" b_ord "∨I"(1) nec_delta "oa-facts:1" "vdash-properties:10")
6167  next AOT_show ‹¬𝒜([O!]a ∨ q0)›
6168      using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_ord_a not_act_q_zero "raa-cor:3" by blast
6169  next AOT_show ‹Δ([O!]a ∨ q0)›
6170    proof (rule not_act_and_pos_delta)
6171      AOT_show ‹¬𝒜([O!]a ∨ q0)›
6172        using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_ord_a not_act_q_zero "raa-cor:3" by blast
6173    next AOT_show ‹◇([O!]a ∨ q0)›
6174        using "KBasic2:2" "&E"(1) "∨I"(2) "≡E"(2) q0_prop by blast
6175    qed
6176  qed
6177  ultimately AOT_obtain F13 where ‹𝒜[F13]b & Δ[F13]b & ¬𝒜[F13]a & Δ[F13]a›
6178    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6179  AOT_hence ‹𝒜[F13]b› and ‹Δ[F13]b› and ‹¬𝒜[F13]a› and ‹Δ[F13]a›
6180    using "&E" by blast+
6181  note props = props this
6182
6183  let ?Π = "«[λy [O!]y ∨ ¬q0]»"
6184  AOT_modally_strict {
6185     AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6186  } note 1 = this
6187  moreover AOT_have ‹𝒜[«?Π»]b & Δ[«?Π»]b & 𝒜[«?Π»]a & ¬Δ[«?Π»]a›
6188  proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6189    AOT_show ‹𝒜([O!]b ∨ ¬q0)›
6190      by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6191  next AOT_show ‹Δ([O!]b ∨ ¬q0)›
6192      by (meson "KBasic:15" b_ord "∨I"(1) nec_delta "oa-facts:1" "vdash-properties:10")
6193  next AOT_show ‹𝒜([O!]a ∨ ¬q0)›
6194      by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6195  next AOT_show ‹¬Δ([O!]a ∨ ¬q0)›
6196    proof(rule act_and_pos_not_not_delta)
6197      AOT_show ‹𝒜([O!]a ∨ ¬q0)›
6198        by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6199    next
6200      AOT_have ‹□¬[O!]a›
6201        using "KBasic2:1" "≡E"(2) not_act_and_pos_delta not_act_ord_a not_delta_ord_a "raa-cor:6" by blast
6202      moreover AOT_have ‹◇q0›
6203        by (meson "&E"(1) q0_prop)
6204      ultimately AOT_have 2: ‹◇(¬[O!]a & q0)›
6205         by (metis "KBasic:16" "&I" "vdash-properties:10")
6206      AOT_show ‹◇¬([O!]a ∨ ¬q0)›
6207      proof (AOT_subst_rev ‹«¬[O!]a & q0»› ‹«¬([O!]a ∨ ¬q0)»›)
6208        AOT_modally_strict {
6209          AOT_show ‹¬[O!]a & q0 ≡ ¬([O!]a ∨ ¬q0)›
6210            by (metis "&I" "&E"(1) "&E"(2) "∨I"(1) "∨I"(2)
6211                      "∨E"(3) "deduction-theorem" "≡I" "raa-cor:3")
6212        }
6213      next
6214        AOT_show ‹◇(¬[O!]a & q0)›
6215          using "2" by blast
6216      qed
6217    qed
6218  qed
6219  ultimately AOT_obtain F14 where ‹𝒜[F14]b & Δ[F14]b & 𝒜[F14]a & ¬Δ[F14]a›
6220    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6221  AOT_hence ‹𝒜[F14]b› and ‹Δ[F14]b› and ‹𝒜[F14]a› and ‹¬Δ[F14]a›
6222    using "&E" by blast+
6223  note props = props this
6224
6225  AOT_have ‹[L]↓›
6226    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
6227  moreover AOT_have ‹𝒜[L]b & Δ[L]b & 𝒜[L]a & Δ[L]a›
6228  proof (safe intro!: "&I")
6229    AOT_show ‹𝒜[L]b›
6230      by (meson nec_L "nec-imp-act" "vdash-properties:10")
6231    next AOT_show ‹Δ[L]b› using nec_L nec_delta by blast
6232    next AOT_show ‹𝒜[L]a› by (meson nec_L "nec-imp-act" "vdash-properties:10")
6233    next AOT_show ‹Δ[L]a› using nec_L nec_delta by blast
6234  qed
6235  ultimately AOT_obtain F15 where ‹𝒜[F15]b & Δ[F15]b & 𝒜[F15]a & Δ[F15]a›
6236    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6237  AOT_hence ‹𝒜[F15]b› and ‹Δ[F15]b› and ‹𝒜[F15]a› and ‹Δ[F15]a›
6238    using "&E" by blast+
6239  note props = props this
6240
6241  show ?thesis
6242    by (rule "∃I"(2)[where β=F0]; rule "∃I"(2)[where β=F1]; rule "∃I"(2)[where β=F2];
6243           rule "∃I"(2)[where β=F3]; rule "∃I"(2)[where β=F4]; rule "∃I"(2)[where β=F5];
6244           rule "∃I"(2)[where β=F6]; rule "∃I"(2)[where β=F7]; rule "∃I"(2)[where β=F8];
6245           rule "∃I"(2)[where β=F9]; rule "∃I"(2)[where β=F10]; rule "∃I"(2)[where β=F11];
6246           rule "∃I"(2)[where β=F12]; rule "∃I"(2)[where β=F13]; rule "∃I"(2)[where β=F14];
6247           rule "∃I"(2)[where β=F15]; safe intro!: "&I")
6248       (match conclusion in "[?v ⊨ [F] ≠ [G]]" for F G ⇒ ‹
6249        match props in A: "[?v ⊨ ¬φ{F}]" for φ ⇒ ‹
6250        match (φ) in "λa . ?p" ⇒ ‹fail› ¦ "λa . a" ⇒ ‹fail› ¦ _ ⇒ ‹
6251        match props in B: "[?v ⊨ φ{G}]" ⇒ ‹
6252        fact "pos-not-equiv-ne:4"[where F=F and G=G and φ=φ, THEN "→E",
6253                                OF "oth-class-taut:4:h"[THEN "≡E"(2)],
6254                                OF "Disjunction Addition"(2)[THEN "→E"],
6255                                OF "&I", OF A, OF B]››››)+
6256qed
6257
6258AOT_theorem "o-objects-exist:1": ‹□∃x O!x›
6259proof(rule RN)
6260  AOT_modally_strict {
6261    AOT_obtain a where ‹◇(E!a & ¬𝒜[E!]a)›
6262      using "∃E"[rotated, OF "qml:4"[axiom_inst, THEN "BF◇"[THEN "→E"]]] by blast
6263    AOT_hence 1: ‹◇E!a› by (metis "KBasic2:3" "&E"(1) "→E")
6264    AOT_have ‹[λx ◇[E!]x]a›
6265    proof (rule "β←C"(1); "cqt:2[lambda]"?)
6266      AOT_show ‹a↓› using "cqt:2[const_var]"[axiom_inst] by blast
6267    next
6268      AOT_show ‹◇E!a› by (fact 1)
6269    qed
6270    AOT_hence ‹O!a› by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
6271    AOT_thus ‹∃x [O!]x› by (rule "∃I")
6272  }
6273qed
6274
6275AOT_theorem "o-objects-exist:2": ‹□∃x A!x›
6276proof (rule RN)
6277  AOT_modally_strict {
6278    AOT_obtain a where ‹[A!]a›
6279      using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
6280    AOT_thus ‹∃x A!x› using "∃I" by blast
6281  }
6282qed
6283
6284AOT_theorem "o-objects-exist:3": ‹□¬∀x O!x›
6285  by (rule RN) (metis (no_types, hide_lams) "∃E" "cqt-orig:1[const_var]" "≡E"(4) "modus-tollens:1" "o-objects-exist:2" "oa-contingent:2" "qml:2"[axiom_inst] "reductio-aa:2")
6286
6287AOT_theorem "o-objects-exist:4": ‹□¬∀x A!x›
6288  by (rule RN) (metis (mono_tags, hide_lams) "∃E" "cqt-orig:1[const_var]" "≡E"(1) "modus-tollens:1" "o-objects-exist:1" "oa-contingent:2" "qml:2"[axiom_inst] "→E")
6289
6290AOT_theorem "o-objects-exist:5": ‹□¬∀x E!x›
6291proof (rule RN; rule "raa-cor:2")
6292  AOT_modally_strict {
6293    AOT_assume ‹∀x E!x›
6294    moreover AOT_obtain a where abs: ‹A!a›
6295      using "o-objects-exist:2"[THEN "qml:2"[axiom_inst, THEN "→E"]] "∃E"[rotated] by blast
6296    ultimately AOT_have ‹E!a› using "∀E" by blast
6297    AOT_hence 1: ‹◇E!a› by (metis "T◇" "→E")
6298    AOT_have ‹[λy ◇E!y]a›
6299    proof (rule "β←C"(1); "cqt:2[lambda]"?)
6300      AOT_show ‹a↓› using "cqt:2[const_var]"[axiom_inst].
6301    next
6302      AOT_show ‹◇E!a› by (fact 1)
6303    qed
6304    AOT_hence ‹O!a›
6305      by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
6306    AOT_hence ‹¬A!a› by (metis "≡E"(1) "oa-contingent:2") 
6307    AOT_thus ‹p & ¬p› for p using abs by (metis "raa-cor:3")
6308  }
6309qed
6310
6311AOT_theorem partition: ‹¬∃x (O!x & A!x)›
6312proof(rule "raa-cor:2")
6313  AOT_assume ‹∃x (O!x & A!x)›
6314  then AOT_obtain a where ‹O!a & A!a› using "∃E"[rotated] by blast
6315  AOT_thus ‹p & ¬p› for p by (metis "&E"(1) "Conjunction Simplification"(2) "≡E"(1) "modus-tollens:1" "oa-contingent:2" "raa-cor:3")
6316qed
6317
6318AOT_define eq_E :: ‹Π› ("'(=E')") "=E": ‹(=E) =df [λxy O!x & O!y & □∀F ([F]x ≡ [F]y)]›
6319
6320syntax "_AOT_eq_E_infix" :: ‹τ ⇒ τ ⇒ φ› (infixl "=E" 50)
6321translations
6322  "_AOT_eq_E_infix κ κ'" == "CONST AOT_exe (CONST eq_E) (CONST Pair κ κ')"
6323(* TODO: try to replace by a simple translations pattern *)
6324print_translation‹
6325AOT_syntax_print_translations
6326[(const_syntax‹AOT_exe›, fn ctxt => fn [
6327  Const ("constAOT_PLM.eq_E", _),
6328  Const (const_syntax‹Pair›, _) $ lhs $ rhs
6329] => Const (syntax_const‹_AOT_eq_E_infix›, dummyT) $ lhs $ rhs)]›
6330
6331text‹Note: Not explicitly mentioned as theorem in PLM.›
6332AOT_theorem "=E[denotes]": ‹[(=E)]↓›
6333  by (rule "=dfI"(2)[OF "=E"]) "cqt:2[lambda]"+
6334
6335AOT_theorem "=E-simple:1": ‹x =E y ≡ (O!x & O!y & □∀F ([F]x ≡ [F]y))›
6336proof -
6337  (* TODO: rethink the product hacks *)
6338  AOT_have 0: ‹«(AOT_term_of_var x,AOT_term_of_var y)»↓›
6339    by (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6340  AOT_have 1: ‹[λxy [O!]x & [O!]y & □∀F ([F]x ≡ [F]y)]↓› by "cqt:2[lambda]"
6341  show ?thesis apply (rule "=dfI"(2)[OF "=E"]; "cqt:2[lambda]"?)
6342    using "beta-C-meta"[THEN "→E", OF 1, unvarify ν1νn, of "(AOT_term_of_var x,AOT_term_of_var y)", OF 0]
6343    by fast
6344qed
6345
6346AOT_theorem "=E-simple:2": ‹x =E y → x = y›
6347proof (rule "→I")
6348  AOT_assume ‹x =E y›
6349  AOT_hence ‹O!x & O!y & □∀F ([F]x ≡ [F]y)› using "=E-simple:1"[THEN "≡E"(1)] by blast
6350  AOT_thus ‹x = y›
6351    using "≡dfI"[OF "identity:1"] "∨I" by blast
6352qed
6353
6354AOT_theorem "id-nec3:1": ‹x =E y ≡ □(x =E y)›
6355proof (rule "≡I"; rule "→I")
6356  AOT_assume ‹x =E y›
6357  AOT_hence ‹O!x & O!y & □∀F ([F]x ≡ [F]y)›
6358    using "=E-simple:1" "≡E" by blast
6359  AOT_hence ‹□O!x & □O!y & □□∀F ([F]x ≡ [F]y)›
6360    by (metis "S5Basic:6" "&I" "&E"(1) "&E"(2) "≡E"(4) "oa-facts:1" "raa-cor:3" "vdash-properties:10")
6361  AOT_hence 1: ‹□(O!x & O!y & □∀F ([F]x ≡ [F]y))›
6362    by (metis "&E"(1) "&E"(2) "≡E"(2) "KBasic:3" "&I")
6363  AOT_show ‹□(x =E y)›
6364    apply (AOT_subst ‹«x =E y»› ‹«O!x & O!y & □∀F ([F]x ≡ [F]y)»›)
6365     using "=E-simple:1" apply presburger
6366    by (simp add: "1")
6367next
6368  AOT_assume ‹□(x =E y)›
6369  AOT_thus ‹x =E y› using "qml:2"[axiom_inst, THEN "→E"] by blast
6370qed
6371
6372AOT_theorem "id-nec3:2": ‹◇(x =E y) ≡ x =E y›
6373  by (meson "RE◇" "S5Basic:2" "id-nec3:1" "≡E"(1) "≡E"(5) "Commutativity of ≡")
6374
6375AOT_theorem "id-nec3:3": ‹◇(x =E y) ≡ □(x =E y)›
6376  by (meson "id-nec3:1" "id-nec3:2" "≡E"(5))
6377
6378syntax "_AOT_non_eq_E" :: ‹Π› ("'(≠E')")
6379translations
6380  (Π) "(≠E)" == (Π) "(=E)-"
6381syntax "_AOT_non_eq_E_infix" :: ‹τ ⇒ τ ⇒ φ› (infixl "≠E" 50)
6382translations
6383 "_AOT_non_eq_E_infix κ κ'" == "CONST AOT_exe (CONST relation_negation (CONST eq_E)) (CONST Pair κ κ')"
6384(* TODO: try replacing be a simple translations pattern *)
6385print_translation‹
6386AOT_syntax_print_translations
6387[(const_syntax‹AOT_exe›, fn ctxt => fn [
6388  Const (const_syntax‹relation_negation›, _) $ Const ("constAOT_PLM.eq_E", _),
6389  Const (const_syntax‹Pair›, _) $ lhs $ rhs
6390] => Const (syntax_const‹_AOT_non_eq_E_infix›, dummyT) $ lhs $ rhs)]›
6391AOT_theorem "thm-neg=E": ‹x ≠E y ≡ ¬(x =E y)›
6392proof -
6393  (* TODO: rethink the product hacks *)
6394  AOT_have 0: ‹«(AOT_term_of_var x,AOT_term_of_var y)»↓›
6395    by (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6396  AOT_have θ: ‹[λx1...x2 ¬(=E)x1...x2]↓› by "cqt:2[lambda]" (* TODO_PLM: convoluted proof in PLM; TODO: product hack *)
6397  AOT_have ‹x ≠E y ≡ [λx1...x2 ¬(=E)x1...x2]xy›
6398    by (rule "=dfI"(1)[OF "df-relation-negation", OF θ])
6399       (meson "oth-class-taut:3:a")
6400  also AOT_have ‹… ≡ ¬(=E)xy›
6401    apply (rule "beta-C-meta"[THEN "→E", unvarify ν1νn])
6402     apply "cqt:2[lambda]"
6403    by (fact 0)
6404  finally show ?thesis.
6405qed
6406
6407AOT_theorem "id-nec4:1": ‹x ≠E y ≡ □(x ≠E y)›
6408proof -
6409  AOT_have ‹x ≠E y ≡ ¬(x =E y)› using "thm-neg=E".
6410  also AOT_have ‹… ≡ ¬◇(x =E y)›
6411    by (meson "id-nec3:2" "≡E"(1) "Commutativity of ≡" "oth-class-taut:4:b")
6412  also AOT_have ‹… ≡ □¬(x =E y)›
6413    by (meson "KBasic2:1" "≡E"(2) "Commutativity of ≡")
6414  also AOT_have ‹… ≡ □(x ≠E y)›
6415    by (AOT_subst_rev "«x ≠E y»" "«¬(x =E y)»")
6416       (auto simp: "thm-neg=E" "oth-class-taut:3:a")
6417  finally show ?thesis.
6418qed
6419
6420AOT_theorem "id-nec4:2": ‹◇(x ≠E y) ≡ (x ≠E y)›
6421  by (meson "RE◇" "S5Basic:2" "id-nec4:1" "≡E"(2) "≡E"(5) "Commutativity of ≡")
6422
6423AOT_theorem "id-nec4:3": ‹◇(x ≠E y) ≡ □(x ≠E y)›
6424  by (meson "id-nec4:1" "id-nec4:2" "≡E"(5))
6425
6426AOT_theorem "id-act2:1": ‹x =E y ≡ 𝒜x =E y›
6427  by (meson "Act-Basic:5" "Act-Sub:2" "RA[2]" "id-nec3:2" "≡E"(1) "≡E"(6))
6428AOT_theorem "id-act2:2": ‹x ≠E y ≡ 𝒜x ≠E y›
6429  by (meson "Act-Basic:5" "Act-Sub:2" "RA[2]" "id-nec4:2" "≡E"(1) "≡E"(6))
6430
6431AOT_theorem "ord=Eequiv:1": ‹O!x → x =E x›
6432proof (rule "→I")
6433  AOT_assume 1: ‹O!x›
6434  AOT_show ‹x =E x›
6435    apply (rule "=dfI"(2)[OF "=E"]) apply "cqt:2[lambda]"
6436    apply (rule "β←C"(1))
6437      apply "cqt:2[lambda]"
6438     apply (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6439    by (simp add: "1" RN "&I" "oth-class-taut:3:a" "universal-cor")
6440qed
6441
6442AOT_theorem "ord=Eequiv:2": ‹x =E y → y =E x›
6443proof(rule CP)
6444  AOT_assume 1: ‹x =E y›
6445  AOT_hence 2: ‹x = y› by (metis "=E-simple:2" "vdash-properties:10") 
6446  AOT_have ‹O!x› using 1 by (meson "&E"(1) "=E-simple:1" "≡E"(1))
6447  AOT_hence ‹x =E x› using "ord=Eequiv:1" "→E" by blast
6448  AOT_thus ‹y =E x› using "rule=E"[rotated, OF 2] by fast
6449qed
6450
6451AOT_theorem "ord=Eequiv:3": ‹(x =E y & y =E z) → x =E z›
6452proof (rule CP)
6453  AOT_assume 1: ‹x =E y & y =E z›
6454  AOT_hence ‹x = y & y = z›
6455    by (metis "&I" "&E"(1) "&E"(2) "=E-simple:2" "vdash-properties:6")
6456  AOT_hence ‹x = z› by (metis "id-eq:3" "vdash-properties:6")
6457  moreover AOT_have ‹x =E x›
6458    using 1[THEN "&E"(1)] "&E"(1) "=E-simple:1" "≡E"(1) "ord=Eequiv:1" "→E" by blast
6459  ultimately AOT_show ‹x =E z›
6460    using "rule=E" by fast
6461qed
6462
6463AOT_theorem "ord-=E=:1": ‹(O!x ∨ O!y) → □(x = y ≡ x =E y)›
6464proof(rule CP)
6465  AOT_assume ‹O!x ∨ O!y›
6466  moreover {
6467    AOT_assume ‹O!x›
6468    AOT_hence ‹□O!x› by (metis "oa-facts:1" "vdash-properties:10")
6469    moreover {
6470      AOT_modally_strict {
6471        AOT_have ‹O!x → (x = y ≡ x =E y)›
6472        proof (rule "→I"; rule "≡I"; rule "→I")
6473          AOT_assume ‹O!x›
6474          AOT_hence ‹x =E x› by (metis "ord=Eequiv:1" "→E")
6475          moreover AOT_assume ‹x = y›
6476          ultimately AOT_show ‹x =E y› using "rule=E" by fast
6477        next
6478          AOT_assume ‹x =E y›
6479          AOT_thus ‹x = y› by (metis "=E-simple:2" "→E")
6480        qed
6481      }
6482      AOT_hence ‹□O!x → □(x = y ≡ x =E y)› by (metis "RM:1")
6483    }
6484    ultimately AOT_have ‹□(x = y ≡ x =E y)› using "→E" by blast
6485  }
6486  moreover {
6487    AOT_assume ‹O!y›
6488    AOT_hence ‹□O!y› by (metis "oa-facts:1" "vdash-properties:10")
6489    moreover {
6490      AOT_modally_strict {
6491        AOT_have ‹O!y → (x = y ≡ x =E y)›
6492        proof (rule "→I"; rule "≡I"; rule "→I")
6493          AOT_assume ‹O!y›
6494          AOT_hence ‹y =E y› by (metis "ord=Eequiv:1" "→E")
6495          moreover AOT_assume ‹x = y›
6496          ultimately AOT_show ‹x =E y› using "rule=E" id_sym by fast
6497        next
6498          AOT_assume ‹x =E y›
6499          AOT_thus ‹x = y› by (metis "=E-simple:2" "→E")
6500        qed
6501      }
6502      AOT_hence ‹□O!y → □(x = y ≡ x =E y)› by (metis "RM:1")
6503    }
6504    ultimately AOT_have ‹□(x = y ≡ x =E y)› using "→E" by blast
6505  }
6506  ultimately AOT_show ‹□(x = y ≡ x =E y)› by (metis "∨E"(3) "raa-cor:1")
6507qed
6508
6509AOT_theorem "ord-=E=:2": ‹O!y → [λx x = y]↓›
6510proof (rule "→I"; rule "safe-ext"[axiom_inst, THEN "→E"]; rule "&I")
6511  AOT_show ‹[λx x =E y]↓› by "cqt:2[lambda]"
6512next
6513  AOT_assume ‹O!y›
6514  AOT_hence 1: ‹□(x = y ≡ x =E y)› for x using "ord-=E=:1" "→E" "∨I" by blast
6515  AOT_have ‹□(x =E y ≡ x = y)› for x
6516    by (AOT_subst ‹«x =E y ≡ x = y»› ‹«x = y ≡ x =E y»›)
6517       (auto simp add: "Commutativity of ≡" 1)
6518  AOT_hence ‹∀x □(x =E y ≡ x = y)› by (rule GEN)
6519  AOT_thus ‹□∀x (x =E y ≡ x = y)› by (rule BF[THEN "→E"])
6520qed
6521
6522
6523AOT_theorem "ord-=E=:3": ‹[λxy O!x & O!y & x = y]↓›
6524proof (rule "safe-ext[2]"[axiom_inst, THEN "→E"]; rule "&I")
6525  AOT_show ‹[λxy O!x & O!y & x =E y]↓› by "cqt:2[lambda]"
6526next
6527  AOT_show ‹□∀x∀y ([O!]x & [O!]y & x =E y ≡ [O!]x & [O!]y & x = y)›
6528  proof (rule RN; rule GEN; rule GEN; rule "≡I"; rule "→I")
6529    AOT_modally_strict {
6530      AOT_show ‹[O!]x & [O!]y & x = y› if ‹[O!]x & [O!]y & x =E y› for x y
6531        by (metis "&I" "&E"(1) "Conjunction Simplification"(2) "=E-simple:2"
6532                  "modus-tollens:1" "raa-cor:1" that)
6533    }
6534  next
6535    AOT_modally_strict {
6536      AOT_show ‹[O!]x & [O!]y & x =E y› if ‹[O!]x & [O!]y & x = y› for x y
6537        apply(safe intro!: "&I")
6538          apply (metis that[THEN "&E"(1), THEN "&E"(1)])
6539         apply (metis that[THEN "&E"(1), THEN "&E"(2)])
6540        using "rule=E"[rotated, OF that[THEN "&E"(2)]]
6541              "ord=Eequiv:1"[THEN "→E", OF that[THEN "&E"(1), THEN "&E"(1)]] by fast
6542    }
6543  qed
6544qed
6545
6546AOT_theorem "ind-nec": ‹∀F ([F]x ≡ [F]y) → □∀F ([F]x ≡ [F]y)›
6547proof(rule "→I")
6548  AOT_assume ‹∀F ([F]x ≡ [F]y)›
6549  moreover AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]↓› by "cqt:2[lambda]"
6550  ultimately AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]x ≡ [λx □∀F ([F]x ≡ [F]y)]y›
6551    using "∀E" by blast
6552  moreover AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]y›
6553    apply (rule "β←C"(1))
6554      apply "cqt:2[lambda]"
6555     apply (fact "cqt:2[const_var]"[axiom_inst])
6556    by (simp add: RN GEN "oth-class-taut:3:a")
6557  ultimately AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]x› using "≡E" by blast
6558  AOT_thus ‹□∀F ([F]x ≡ [F]y)›
6559    using "β→C"(1) by blast
6560qed
6561
6562AOT_theorem "ord=E:1": ‹(O!x & O!y) → (∀F ([F]x ≡ [F]y) → x =E y)›
6563proof (rule "→I"; rule "→I")
6564  AOT_assume ‹∀F ([F]x ≡ [F]y)›
6565  AOT_hence ‹□∀F ([F]x ≡ [F]y)›
6566    using "ind-nec"[THEN "→E"] by blast
6567  moreover AOT_assume ‹O!x & O!y›
6568  ultimately AOT_have ‹O!x & O!y & □∀F ([F]x ≡ [F]y)›
6569    using "&I" by blast
6570  AOT_thus ‹x =E y› using "=E-simple:1"[THEN "≡E"(2)] by blast
6571qed
6572
6573AOT_theorem "ord=E:2": ‹(O!x & O!y) → (∀F ([F]x ≡ [F]y) → x = y)›
6574proof (rule "→I"; rule "→I")
6575  AOT_assume ‹O!x & O!y›
6576  moreover AOT_assume ‹∀F ([F]x ≡ [F]y)›
6577  ultimately AOT_have ‹x =E y›
6578    using "ord=E:1" "→E" by blast
6579  AOT_thus ‹x = y› using "=E-simple:2"[THEN "→E"] by blast
6580qed
6581
6582AOT_theorem "ord=E2:1": ‹(O!x & O!y) → (x ≠ y ≡ [λz z =E x] ≠ [λz z =E y])›
6583proof (rule "→I"; rule "≡I"; rule "→I"; rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
6584  AOT_assume 0: ‹O!x & O!y›
6585  AOT_assume ‹x ≠ y›
6586  AOT_hence 1: ‹¬(x = y)› using "≡dfE"[OF "=-infix"] by blast
6587  AOT_assume ‹[λz z =E x] = [λz z =E y]›
6588  moreover AOT_have ‹[λz z =E x]x›
6589    apply (rule "β←C"(1))
6590      apply "cqt:2[lambda]"
6591     apply (fact "cqt:2[const_var]"[axiom_inst])
6592    using "ord=Eequiv:1"[THEN "→E", OF 0[THEN "&E"(1)]].
6593  ultimately AOT_have ‹[λz z =E y]x› using "rule=E" by fast
6594  AOT_hence ‹x =E y› using "β→C"(1) by blast
6595  AOT_hence ‹x = y› by (metis "=E-simple:2" "vdash-properties:6")
6596  AOT_thus ‹x = y & ¬(x = y)› using 1 "&I" by blast
6597next
6598  AOT_assume ‹[λz z =E x] ≠ [λz z =E y]›
6599  AOT_hence 0: ‹¬([λz z =E x] = [λz z =E y])› using "≡dfE"[OF "=-infix"] by blast
6600  AOT_have ‹[λz z =E x]↓› by "cqt:2[lambda]"
6601  AOT_hence ‹[λz z =E x] = [λz z =E x]›
6602    by (metis "rule=I:1")
6603  moreover AOT_assume ‹x = y›
6604  ultimately AOT_have ‹[λz z =E x] = [λz z =E y]›
6605    using "rule=E" by fast
6606  AOT_thus ‹[λz z =E x] = [λz z =E y] & ¬([λz z =E x] = [λz z =E y])›
6607    using 0 "&I" by blast
6608qed
6609
6610AOT_theorem "ord=E2:2": ‹(O!x & O!y) → (x ≠ y ≡ [λz z = x] ≠ [λz z = y])›
6611proof (rule "→I"; rule "≡I"; rule "→I"; rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
6612  AOT_assume 0: ‹O!x & O!y›
6613  AOT_assume ‹x ≠ y›
6614  AOT_hence 1: ‹¬(x = y)› using "≡dfE"[OF "=-infix"] by blast
6615  AOT_assume ‹[λz z = x] = [λz z = y]›
6616  moreover AOT_have ‹[λz z = x]x›
6617    apply (rule "β←C"(1))
6618    apply (fact "ord-=E=:2"[THEN "→E", OF 0[THEN "&E"(1)]])
6619     apply (fact "cqt:2[const_var]"[axiom_inst])
6620    by (simp add: "id-eq:1")
6621  ultimately AOT_have ‹[λz z = y]x› using "rule=E" by fast
6622  AOT_hence ‹x = y› using "β→C"(1) by blast
6623  AOT_thus ‹x = y & ¬(x = y)› using 1 "&I" by blast
6624next
6625  AOT_assume 0: ‹O!x & O!y›
6626  AOT_assume ‹[λz z = x] ≠ [λz z = y]›
6627  AOT_hence 1: ‹¬([λz z = x] = [λz z = y])› using "≡dfE"[OF "=-infix"] by blast
6628  AOT_have ‹[λz z = x]↓› by (fact "ord-=E=:2"[THEN "→E", OF 0[THEN "&E"(1)]])
6629  AOT_hence ‹[λz z = x] = [λz z = x]›
6630    by (metis "rule=I:1")
6631  moreover AOT_assume ‹x = y›
6632  ultimately AOT_have ‹[λz z = x] = [λz z = y]›
6633    using "rule=E" by fast
6634  AOT_thus ‹[λz z = x] = [λz z = y] & ¬([λz z = x] = [λz z = y])›
6635    using 1 "&I" by blast
6636qed
6637
6638AOT_theorem ordnecfail: ‹O!x → □¬∃F x[F]›
6639  by (meson "RM:1" "deduction-theorem" nocoder "oa-facts:1" "vdash-properties:10" "vdash-properties:1[2]")
6640
6641AOT_theorem "ab-obey:1": ‹(A!x & A!y) → (∀F (x[F] ≡ y[F]) → x = y)›
6642proof (rule "→I"; rule "→I")
6643  AOT_assume 1: ‹A!x & A!y›
6644  AOT_assume ‹∀F (x[F] ≡ y[F])›
6645  AOT_hence ‹x[F] ≡ y[F]› for F using "∀E" by blast
6646  AOT_hence ‹□(x[F] ≡ y[F])› for F by (metis "en-eq:6[1]" "≡E"(1))
6647  AOT_hence ‹∀F □(x[F] ≡ y[F])› by (rule GEN)
6648  AOT_hence ‹□∀F (x[F] ≡ y[F])› by (rule BF[THEN "→E"])
6649  AOT_thus ‹x = y›
6650    using "≡dfI"[OF "identity:1", OF "∨I"(2)] 1 "&I" by blast
6651qed
6652
6653AOT_theorem "ab-obey:2": ‹(∃F (x[F] & ¬y[F]) ∨ ∃F (y[F] & ¬x[F])) → x ≠ y›
6654proof (rule "→I"; rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
6655  AOT_assume 1: ‹x = y›
6656  AOT_assume ‹∃F (x[F] & ¬y[F]) ∨ ∃F (y[F] & ¬x[F])›
6657  moreover {
6658    AOT_assume ‹∃F (x[F] & ¬y[F])›
6659    then AOT_obtain F where ‹x[F] & ¬y[F]› using "∃E"[rotated] by blast
6660    moreover AOT_have ‹y[F]› using calculation[THEN "&E"(1)] 1 "rule=E" by fast
6661    ultimately AOT_have ‹p & ¬p› for p by (metis "Conjunction Simplification"(2) "modus-tollens:2" "raa-cor:3")
6662  }
6663  moreover {
6664    AOT_assume ‹∃F (y[F] & ¬x[F])›
6665    then AOT_obtain F where ‹y[F] & ¬x[F]› using "∃E"[rotated] by blast
6666    moreover AOT_have ‹¬y[F]› using calculation[THEN "&E"(2)] 1 "rule=E" by fast
6667    ultimately AOT_have ‹p & ¬p› for p by (metis "Conjunction Simplification"(1) "modus-tollens:1" "raa-cor:3")
6668  }
6669  ultimately AOT_show ‹p & ¬p› for p by (metis "∨E"(3) "raa-cor:1")
6670qed
6671
6672AOT_theorem "encoders-are-abstract": ‹∃F x[F] → A!x›
6673  by (meson "deduction-theorem" "≡E"(2) "modus-tollens:2" nocoder
6674            "oa-contingent:3" "vdash-properties:1[2]")
6675
6676AOT_theorem "denote=:1": ‹∀H∃x x[H]›
6677  by (rule GEN; rule "existence:2[1]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6678
6679AOT_theorem "denote=:2": ‹∀G∃x1...∃xn x1...xn[H]›
6680  by (rule GEN; rule "existence:2"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6681
6682AOT_theorem "denote=:2[2]": ‹∀G∃x1∃x2 x1x2[H]›
6683  by (rule GEN; rule "existence:2[2]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6684
6685AOT_theorem "denote=:2[3]": ‹∀G∃x1∃x2∃x3 x1x2x3[H]›
6686  by (rule GEN; rule "existence:2[3]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6687
6688AOT_theorem "denote=:2[4]": ‹∀G∃x1∃x2∃x3∃x4 x1x2x3x4[H]›
6689  by (rule GEN; rule "existence:2[4]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6690
6691AOT_theorem "denote=:3": ‹∃x x[Π] ≡ ∃H (H = Π)›
6692  using "existence:2[1]" "free-thms:1" "≡E"(2) "≡E"(5) "Commutativity of ≡" "≡Df" by blast
6693
6694AOT_theorem "denote=:4": ‹(∃x1...∃xn x1...xn[Π]) ≡ ∃H (H = Π)›
6695  using "existence:2" "free-thms:1" "≡E"(6) "≡Df" by blast
6696
6697AOT_theorem "denote=:4[2]": ‹(∃x1∃x2 x1x2[Π]) ≡ ∃H (H = Π)›
6698  using "existence:2[2]" "free-thms:1" "≡E"(6) "≡Df" by blast
6699
6700AOT_theorem "denote=:4[3]": ‹(∃x1∃x2∃x3 x1x2x3[Π]) ≡ ∃H (H = Π)›
6701  using "existence:2[3]" "free-thms:1" "≡E"(6) "≡Df" by blast
6702
6703AOT_theorem "denote=:4[4]": ‹(∃x1∃x2∃x3∃x4 x1x2x3x4[Π]) ≡ ∃H (H = Π)›
6704  using "existence:2[4]" "free-thms:1" "≡E"(6) "≡Df" by blast
6705
6706AOT_theorem "A-objects!": ‹∃!x (A!x & ∀F (x[F] ≡ φ{F}))›
6707proof (rule "uniqueness:1"[THEN "≡dfI"])
6708  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ φ{F})›
6709    using "A-objects"[axiom_inst] "∃E"[rotated] by blast
6710  AOT_have ‹(A!β & ∀F (β[F] ≡ φ{F})) → β = a› for β
6711  proof (rule "→I")
6712    AOT_assume β_prop: ‹[A!]β & ∀F (β[F] ≡ φ{F})›
6713    AOT_hence ‹β[F] ≡ φ{F}› for F using "∀E" "&E" by blast
6714    AOT_hence ‹β[F] ≡ a[F]› for F
6715      using a_prop[THEN "&E"(2)] "∀E" "≡E"(2) "≡E"(5) "Commutativity of ≡" by fast
6716    AOT_hence ‹∀F (β[F] ≡ a[F])› by (rule GEN)
6717    AOT_thus ‹β = a›
6718      using "ab-obey:1"[THEN "→E", OF "&I"[OF β_prop[THEN "&E"(1)], OF a_prop[THEN "&E"(1)]], THEN "→E"] by blast
6719  qed
6720  AOT_hence ‹∀β ((A!β & ∀F (β[F] ≡ φ{F})) → β = a)› by (rule GEN)
6721  AOT_thus ‹∃α ([A!]α & ∀F (α[F] ≡ φ{F}) & ∀β ([A!]β & ∀F (β[F] ≡ φ{F}) → β = α)) ›
6722    using "∃I" using a_prop "&I" by fast
6723qed
6724
6725AOT_theorem "obj-oth:1": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y))›
6726  using "A-objects!" by fast
6727
6728AOT_theorem "obj-oth:2": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y & [F]z))›
6729  using "A-objects!" by fast
6730
6731AOT_theorem "obj-oth:3": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y ∨ [F]z))›
6732  using "A-objects!" by fast
6733
6734AOT_theorem "obj-oth:4": ‹∃!x (A!x & ∀F (x[F] ≡ □[F]y))›
6735  using "A-objects!" by fast
6736
6737AOT_theorem "obj-oth:5": ‹∃!x (A!x & ∀F (x[F] ≡ F = G))›
6738  using "A-objects!" by fast
6739
6740AOT_theorem "obj-oth:6": ‹∃!x (A!x & ∀F (x[F] ≡ □∀y([G]y → [F]y)))›
6741  using "A-objects!" by fast
6742
6743AOT_theorem "A-descriptions": ‹ιx (A!x & ∀F (x[F] ≡ φ{F}))↓›
6744  by (rule "A-Exists:2"[THEN "≡E"(2)]; rule "RA[2]"; rule "A-objects!")
6745
6746AOT_act_theorem "thm-can-terms2": ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F})) → (A!y & ∀F (y[F] ≡ φ{F}))›
6747  using "y-in:2" by blast
6748
6749AOT_theorem "can-ab2": ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F})) →  A!y›
6750proof(rule "→I")
6751  AOT_assume ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F}))›
6752  AOT_hence ‹𝒜(A!y & ∀F (y[F] ≡ φ{F}))›
6753    using "actual-desc:2"[THEN "→E"] by blast
6754  AOT_hence ‹𝒜A!y› by (metis "Act-Basic:2" "&E"(1) "≡E"(1))
6755  AOT_thus ‹A!y› by (metis "≡E"(2) "oa-facts:8")
6756qed
6757
6758AOT_act_theorem "desc-encode": ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ φ{G}›
6759proof -
6760  AOT_have ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))↓›
6761    by (simp add: "A-descriptions")
6762  AOT_hence ‹A!ιx(A!x & ∀F (x[F] ≡ φ{F})) & ∀F (ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6763    using "y-in:3"[THEN "→E"] by blast
6764  AOT_thus ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ φ{G}›
6765    using "&E" "∀E" by blast
6766qed
6767
6768AOT_theorem "desc-nec-encode": ‹ιx (A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ 𝒜φ{G}›
6769proof -
6770  AOT_have 0: ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))↓›
6771    by (simp add: "A-descriptions")
6772  AOT_hence ‹𝒜(A!ιx(A!x & ∀F (x[F] ≡ φ{F})) & ∀F (ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}))›
6773    using "actual-desc:4"[THEN "→E"] by blast
6774  AOT_hence ‹𝒜∀F (ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6775    using "Act-Basic:2" "&E"(2) "≡E"(1) by blast
6776  AOT_hence ‹∀F 𝒜(ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6777    using "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]" by blast
6778  AOT_hence ‹𝒜(ιx(A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ φ{G})›
6779    using "∀E" by blast
6780  AOT_hence ‹𝒜ιx(A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ 𝒜φ{G}›
6781    using "Act-Basic:5" "≡E"(1) by blast
6782  AOT_thus ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ 𝒜φ{G}›
6783    using "en-eq:10[1]"[unvarify x1, OF 0] "≡E"(6) by blast
6784qed
6785
6786AOT_theorem "Box-desc-encode:1": ‹□φ{G} → ιx(A!x & ∀F (x[F] ≡ φ{G}))[G]›
6787  by (rule "→I"; rule "desc-nec-encode"[THEN "≡E"(2)])
6788     (meson "nec-imp-act" "vdash-properties:10")
6789
6790AOT_theorem "Box-desc-encode:2": ‹□φ{G} → □(ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})›
6791proof(rule CP)
6792  AOT_assume ‹□φ{G}›
6793  AOT_hence ‹□□φ{G}› by (metis "S5Basic:6" "≡E"(1))
6794  moreover AOT_have ‹□□φ{G} → □(ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})›
6795  proof (rule RM; rule "→I")
6796    AOT_modally_strict {
6797      AOT_assume 1: ‹□φ{G}›
6798      AOT_hence ‹ιx(A!x & ∀F (x[F] ≡ φ{G}))[G]› using "Box-desc-encode:1" "→E" by blast
6799      moreover AOT_have ‹φ{G}› using 1 by (meson "qml:2" "vdash-properties:10" "vdash-properties:1[2]")
6800      ultimately AOT_show ‹ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G}›
6801        using "deduction-theorem" "≡I" by simp
6802    }
6803  qed
6804  ultimately AOT_show ‹□(ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})› using "→E" by blast
6805qed
6806
6807definition rigid_condition where ‹rigid_condition φ ≡ ∀v . [v ⊨ ∀α (φ{α} → □φ{α})]›
6808syntax rigid_condition :: ‹id_position ⇒ AOT_prop› ("RIGID'_CONDITION'(_')")
6809
6810AOT_theorem "strict-can:1[E]": assumes ‹RIGID_CONDITION(φ)›
6811  shows ‹∀α (φ{α} → □φ{α})›
6812  using assms[unfolded rigid_condition_def] by auto
6813
6814AOT_theorem "strict-can:1[I]":
6815  assumes ‹⊢□ ∀α (φ{α} → □φ{α})›
6816  shows ‹RIGID_CONDITION(φ)›
6817  using assms rigid_condition_def by auto
6818
6819AOT_theorem "box-phi-a:1": assumes ‹RIGID_CONDITION(φ)›
6820  shows ‹(A!x  & ∀F (x[F] ≡ φ{F})) → □(A!x & ∀F (x[F] ≡ φ{F}))›
6821proof (rule "→I")
6822  AOT_assume a: ‹A!x & ∀F (x[F] ≡ φ{F})›
6823  AOT_hence b: ‹□A!x› by (metis "Conjunction Simplification"(1) "oa-facts:2" "vdash-properties:10")
6824  AOT_have ‹x[F] ≡ φ{F}› for F using a[THEN "&E"(2)] "∀E" by blast
6825  moreover AOT_have ‹□(x[F] → □x[F])› for F by (meson "pre-en-eq:1[1]" RN)
6826  moreover AOT_have ‹□(φ{F} → □φ{F})› for F using RN "strict-can:1[E]"[OF assms] "∀E" by blast
6827  ultimately AOT_have ‹□(x[F] ≡ φ{F})› for F
6828    by (metis "&I" "sc-eq-box-box:5" "vdash-properties:6")
6829  AOT_hence ‹∀F □(x[F] ≡ φ{F})› by (rule GEN)
6830  AOT_hence ‹□∀F (x[F] ≡ φ{F})› by (rule BF[THEN "→E"])
6831  AOT_thus ‹□([A!]x & ∀F (x[F] ≡ φ{F}))›
6832    using b "KBasic:3" "≡S"(1) "≡E"(2) by blast
6833qed
6834
6835AOT_theorem "box-phi-a:2": assumes ‹RIGID_CONDITION(φ)›
6836  shows ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F})) → (A!y & ∀F (y[F] ≡ φ{F}))›
6837proof(rule "→I")
6838  AOT_assume ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F}))›
6839  AOT_hence ‹𝒜(A!y & ∀F (y[F] ≡ φ{F}))› using "actual-desc:2"[THEN "→E"] by fast
6840  AOT_hence abs: ‹𝒜A!y› and ‹𝒜∀F (y[F] ≡ φ{F})›
6841    using "Act-Basic:2" "&E" "≡E"(1) by blast+
6842  AOT_hence ‹∀F 𝒜(y[F] ≡ φ{F})› by (metis "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]")
6843  AOT_hence ‹𝒜(y[F] ≡ φ{F})› for F using "∀E" by blast
6844  AOT_hence ‹𝒜y[F] ≡ 𝒜φ{F}› for F by (metis "Act-Basic:5" "≡E"(1)) 
6845  AOT_hence ‹y[F] ≡ φ{F}› for F
6846    using "sc-eq-fur:2"[THEN "→E", OF "strict-can:1[E]"[OF assms, THEN "∀E"(2)[where β=F], THEN RN]]
6847    by (metis "en-eq:10[1]" "≡E"(6))
6848  AOT_hence ‹∀F (y[F] ≡ φ{F})› by (rule GEN)
6849  AOT_thus ‹[A!]y & ∀F (y[F] ≡ φ{F})› using abs "&I" "≡E"(2) "oa-facts:8" by blast
6850qed
6851
6852AOT_theorem "box-phi-a:3": assumes ‹RIGID_CONDITION(φ)›
6853  shows ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ φ{G}›
6854  using "desc-nec-encode" 
6855    "sc-eq-fur:2"[THEN "→E", OF "strict-can:1[E]"[OF assms, THEN "∀E"(2)[where β=G], THEN RN]]
6856    "≡E"(5) by blast
6857
6858AOT_define Null :: ‹τ ⇒ φ› ("Null'(_')") 
6859  "df-null-uni:1": ‹Null(x) ≡df A!x & ¬∃F x[F]›
6860
6861AOT_define Universal :: ‹τ ⇒ φ› ("Universal'(_')")
6862  "df-null-uni:2": ‹Universal(x) ≡df A!x & ∀F x[F]›
6863
6864AOT_theorem "null-uni-uniq:1": ‹∃!x Null(x)›
6865proof (rule "uniqueness:1"[THEN "≡dfI"])
6866  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ¬(F = F))›
6867    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
6868  AOT_have a_null: ‹¬a[F]› for F
6869  proof (rule "raa-cor:2")
6870    AOT_assume ‹a[F]›
6871    AOT_hence ‹¬(F = F)› using a_prop[THEN "&E"(2)] "∀E" "≡E" by blast
6872    AOT_hence ‹F = F & ¬(F = F)› by (metis "id-eq:1" "raa-cor:3")
6873    AOT_thus ‹p & ¬p› for p  by (metis "raa-cor:1")
6874  qed
6875  AOT_have ‹Null(a) & ∀β (Null(β) → β = a)›
6876  proof (rule "&I")
6877    AOT_have ‹¬∃F a[F]› using a_null by (metis "instantiation" "reductio-aa:1")
6878    AOT_thus ‹Null(a)›
6879      using "df-null-uni:1"[THEN "≡dfI"] a_prop[THEN "&E"(1)] "&I" by metis
6880  next
6881    AOT_show ‹∀β (Null(β) → β = a)›
6882    proof (rule GEN; rule "→I")
6883      fix β
6884      AOT_assume a: ‹Null(β)›
6885      AOT_hence ‹¬∃F β[F]›
6886        using "df-null-uni:1"[THEN "≡dfE"] "&E" by blast
6887      AOT_hence β_null: ‹¬β[F]› for F by (metis "existential:2[const_var]" "reductio-aa:1")
6888      AOT_have ‹∀F (β[F] ≡ a[F])›
6889        apply (rule GEN; rule "≡I"; rule CP)
6890        using "raa-cor:3" β_null a_null by blast+
6891      moreover AOT_have ‹A!β› using a "df-null-uni:1"[THEN "≡dfE"] "&E" by blast
6892      ultimately AOT_show ‹β = a›
6893        using a_prop[THEN "&E"(1)] "ab-obey:1"[THEN "→E", THEN "→E"] "&I" by blast
6894    qed
6895  qed
6896  AOT_thus ‹∃α (Null(α) & ∀β (Null(β) → β = α))› using "∃I"(2) by fast
6897qed
6898
6899AOT_theorem "null-uni-uniq:2": ‹∃!x Universal(x)›
6900proof (rule "uniqueness:1"[THEN "≡dfI"])
6901  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ F = F)›
6902    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
6903  AOT_hence aF: ‹a[F]› for F using "&E" "∀E" "≡E" "id-eq:1" by fast
6904  AOT_hence ‹Universal(a)›
6905    using "df-null-uni:2"[THEN "≡dfI"] "&I" a_prop[THEN "&E"(1)] GEN by blast
6906  moreover AOT_have ‹∀β (Universal(β) → β = a)›
6907  proof (rule GEN; rule "→I")
6908    fix β
6909    AOT_assume ‹Universal(β)›
6910    AOT_hence abs_β: ‹A!β› and ‹β[F]› for F using "df-null-uni:2"[THEN "≡dfE"] "&E" "∀E" by blast+
6911    AOT_hence ‹β[F] ≡ a[F]› for F using aF by (metis "deduction-theorem" "≡I")
6912    AOT_hence ‹∀F (β[F] ≡ a[F])› by (rule GEN)
6913    AOT_thus ‹β = a›
6914      using a_prop[THEN "&E"(1)] "ab-obey:1"[THEN "→E", THEN "→E"] "&I" abs_β by blast
6915  qed
6916  ultimately AOT_show ‹∃α (Universal(α) & ∀β (Universal(β) → β = α))›
6917    using "&I" "∃I" by fast
6918qed
6919
6920AOT_theorem "null-uni-uniq:3": ‹ιx Null(x)↓›
6921  using "A-Exists:2" "RA[2]" "≡E"(2) "null-uni-uniq:1" by blast
6922
6923AOT_theorem "null-uni-uniq:4": ‹ιx Universal(x)↓›
6924  using "A-Exists:2" "RA[2]" "≡E"(2) "null-uni-uniq:2" by blast
6925
6926AOT_define Null_object :: ‹κs› (‹a∅›)
6927  "df-null-uni-terms:1": ‹a∅ =df ιx Null(x)›
6928
6929AOT_define Universal_object :: ‹κs› (‹aV›)
6930  "df-null-uni-terms:2": ‹aV =df ιx Universal(x)›
6931
6932AOT_theorem "null-uni-facts:1": ‹Null(x) → □Null(x)›
6933proof (rule "→I")
6934  AOT_assume ‹Null(x)›
6935  AOT_hence x_abs: ‹A!x› and x_null: ‹¬∃F x[F]›
6936    using "df-null-uni:1"[THEN "≡dfE"] "&E" by blast+
6937  AOT_have ‹¬x[F]› for F using x_null
6938    using "existential:2[const_var]" "reductio-aa:1"
6939    by metis
6940  AOT_hence ‹□¬x[F]› for F by (metis "en-eq:7[1]" "≡E"(1))
6941  AOT_hence ‹∀F □¬x[F]› by (rule GEN)
6942  AOT_hence ‹□∀F ¬x[F]› by (rule BF[THEN "→E"])
6943  moreover AOT_have ‹□∀F ¬x[F] → □¬∃F x[F]›
6944    apply (rule RM)
6945    by (metis (full_types) "instantiation" "cqt:2[const_var]" "deduction-theorem"
6946                           "reductio-aa:1" "rule-ui:1" "vdash-properties:1[2]")
6947  ultimately AOT_have ‹□¬∃F x[F]›
6948    by (metis "→E")
6949  moreover AOT_have ‹□A!x› using x_abs
6950    using "oa-facts:2" "vdash-properties:10" by blast
6951  ultimately AOT_have r: ‹□(A!x & ¬∃F x[F])›
6952    by (metis "KBasic:3" "&I" "≡E"(3) "raa-cor:3")
6953  AOT_show ‹□Null(x)›
6954    by (AOT_subst "«Null(x)»" "«A!x & ¬∃F x[F]»")
6955       (auto simp: "df-null-uni:1" "≡Df" r)
6956qed  
6957
6958AOT_theorem "null-uni-facts:2": ‹Universal(x) → □Universal(x)›
6959proof (rule "→I")
6960  AOT_assume ‹Universal(x)›
6961  AOT_hence x_abs: ‹A!x› and x_univ: ‹∀F x[F]›
6962    using "df-null-uni:2"[THEN "≡dfE"] "&E" by blast+
6963  AOT_have ‹x[F]› for F using x_univ "∀E" by blast
6964  AOT_hence ‹□x[F]› for F by (metis "en-eq:2[1]" "≡E"(1))
6965  AOT_hence ‹∀F □x[F]› by (rule GEN)
6966  AOT_hence ‹□∀F x[F]› by (rule BF[THEN "→E"])
6967  moreover AOT_have ‹□A!x› using x_abs
6968    using "oa-facts:2" "vdash-properties:10" by blast
6969  ultimately AOT_have r: ‹□(A!x & ∀F x[F])›
6970    by (metis "KBasic:3" "&I" "≡E"(3) "raa-cor:3")
6971  AOT_show ‹□Universal(x)›
6972    by (AOT_subst "«Universal(x)»" "«A!x & ∀F x[F]»")
6973       (auto simp add: "df-null-uni:2" "≡Df" r)
6974qed
6975
6976AOT_theorem "null-uni-facts:3": ‹Null(a∅)›
6977  apply (rule "=dfI"(2)[OF "df-null-uni-terms:1"])
6978   apply (simp add: "null-uni-uniq:3")
6979  using "actual-desc:4"[THEN "→E", OF "null-uni-uniq:3"]
6980    "sc-eq-fur:2"[THEN "→E", OF "null-uni-facts:1"[unvarify x, THEN RN, OF "null-uni-uniq:3"], THEN "≡E"(1)]
6981  by blast
6982
6983AOT_theorem "null-uni-facts:4": ‹Universal(aV)›
6984  apply (rule "=dfI"(2)[OF "df-null-uni-terms:2"])
6985   apply (simp add: "null-uni-uniq:4")
6986  using "actual-desc:4"[THEN "→E", OF "null-uni-uniq:4"]
6987    "sc-eq-fur:2"[THEN "→E", OF "null-uni-facts:2"[unvarify x, THEN RN, OF "null-uni-uniq:4"], THEN "≡E"(1)]
6988  by blast
6989
6990AOT_theorem "null-uni-facts:5": ‹a∅ ≠ aV›
6991proof (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"];
6992    rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"];
6993    rule "≡dfI"[OF "=-infix"];
6994    rule "raa-cor:2")
6995  AOT_obtain x where nullx: ‹Null(x)›
6996    by (metis "instantiation" "df-null-uni-terms:1" "existential:1" "null-uni-facts:3"
6997              "null-uni-uniq:3" "rule-id-def:2:b[zero]")
6998  AOT_hence act_null: ‹𝒜Null(x)› by (metis "nec-imp-act" "null-uni-facts:1" "vdash-properties:10")
6999  AOT_assume ‹ιx Null(x) = ιx Universal(x)›
7000  AOT_hence ‹𝒜∀x(Null(x) ≡ Universal(x))›
7001    using "actual-desc:5"[THEN "→E"] by blast
7002  AOT_hence ‹∀x 𝒜(Null(x) ≡ Universal(x))›
7003    by (metis "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]")
7004  AOT_hence ‹𝒜Null(x) ≡ 𝒜Universal(x)›
7005    using "Act-Basic:5" "≡E"(1) "rule-ui:3" by blast
7006  AOT_hence ‹𝒜Universal(x)› using act_null "≡E" by blast
7007  AOT_hence ‹Universal(x)› by (metis RN "≡E"(1) "null-uni-facts:2" "sc-eq-fur:2" "vdash-properties:10")
7008  AOT_hence ‹∀F x[F]› using "≡dfE"[OF "df-null-uni:2"] "&E" by metis
7009  moreover AOT_have ‹¬∃F x[F]› using nullx "≡dfE"[OF "df-null-uni:1"] "&E" by metis
7010  ultimately AOT_show ‹p & ¬p› for p by (metis "cqt-further:1" "raa-cor:3" "vdash-properties:10")
7011qed
7012
7013AOT_theorem "null-uni-facts:6": ‹a∅ = ιx(A!x & ∀F (x[F] ≡ F ≠ F))›
7014proof (rule "ab-obey:1"[unvarify x y, THEN "→E", THEN "→E"])
7015  AOT_show ‹ιx([A!]x & ∀F (x[F] ≡ F ≠ F))↓›
7016    by (simp add: "A-descriptions")
7017next
7018  AOT_show ‹a∅↓›
7019    by (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"])
7020       (simp add: "null-uni-uniq:3")
7021next
7022  AOT_have ‹ιx([A!]x & ∀F (x[F] ≡ F ≠ F))↓›
7023    by (simp add: "A-descriptions")
7024  AOT_hence 1: ‹ιx([A!]x & ∀F (x[F] ≡ F ≠ F)) = ιx([A!]x & ∀F (x[F] ≡ F ≠ F))›
7025    using "rule=I:1" by blast
7026  AOT_show ‹[A!]a∅ & [A!]ιx([A!]x & ∀F (x[F] ≡ F ≠ F))›
7027    apply (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"]; rule "&I")
7028    apply (meson "≡dfE" "Conjunction Simplification"(1) "df-null-uni:1" "df-null-uni-terms:1" "null-uni-facts:3" "null-uni-uniq:3" "rule-id-def:2:a[zero]" "vdash-properties:10")
7029    using "can-ab2"[unvarify y, OF "A-descriptions", THEN "→E", OF 1].
7030next
7031  AOT_show ‹∀F (a∅[F] ≡ ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F])›
7032  proof (rule GEN)
7033    fix F
7034    AOT_have ‹¬a∅[F]›
7035      by (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"])
7036         (metis (no_types, lifting) "≡dfE" "&E"(2) "∨I"(2) "∨E"(3)
7037                "df-null-uni:1" "df-null-uni-terms:1" "existential:2[const_var]" "null-uni-facts:3"
7038                "raa-cor:2" "rule-id-def:2:a[zero]" "russell-axiom[enc,1].ψ_denotes_asm")
7039    moreover AOT_have ‹¬ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7040    proof(rule "raa-cor:2")
7041      AOT_assume 0: ‹ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7042      AOT_hence ‹𝒜(F ≠ F)› using "desc-nec-encode"[THEN "≡E"(1), OF 0] by blast
7043      moreover AOT_have ‹¬𝒜(F ≠ F)›
7044        using "≡dfE" "id-act:2" "id-eq:1" "≡E"(2) "=-infix" "raa-cor:3" by blast
7045      ultimately AOT_show ‹𝒜(F ≠ F) & ¬𝒜(F ≠ F)› by (rule "&I")
7046    qed
7047    ultimately AOT_show ‹a∅[F] ≡ ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7048      using "deduction-theorem" "≡I" "raa-cor:4" by blast
7049  qed
7050qed
7051
7052AOT_theorem "null-uni-facts:7": ‹aV = ιx(A!x & ∀F (x[F] ≡ F = F))›
7053proof (rule "ab-obey:1"[unvarify x y, THEN "→E", THEN "→E"])
7054  AOT_show ‹ιx([A!]x & ∀F (x[F] ≡ F = F))↓›
7055    by (simp add: "A-descriptions")
7056next
7057  AOT_show ‹aV↓›
7058    by (rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"])
7059       (simp add: "null-uni-uniq:4")
7060next
7061  AOT_have ‹ιx([A!]x & ∀F (x[F] ≡ F = F))↓›
7062    by (simp add: "A-descriptions")
7063  AOT_hence 1: ‹ιx([A!]x & ∀F (x[F] ≡ F = F)) = ιx([A!]x & ∀F (x[F] ≡ F = F))›
7064    using "rule=I:1" by blast
7065  AOT_show ‹[A!]aV & [A!]ιx([A!]x & ∀F (x[F] ≡ F = F))›
7066    apply (rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"]; rule "&I")
7067    apply (meson "≡dfE" "Conjunction Simplification"(1) "df-null-uni:2" "df-null-uni-terms:2" "null-uni-facts:4" "null-uni-uniq:4" "rule-id-def:2:a[zero]" "vdash-properties:10")
7068    using "can-ab2"[unvarify y, OF "A-descriptions", THEN "→E", OF 1].
7069next
7070  AOT_show ‹∀F (aV[F] ≡ ιx([A!]x & ∀F (x[F] ≡ F = F))[F])›
7071  proof (rule GEN)
7072    fix F
7073    AOT_have ‹aV[F]›
7074      apply (rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"])
7075      using "≡dfE" "&E"(2) "df-null-uni:2" "df-null-uni-terms:2" "null-uni-facts:4" "null-uni-uniq:4" "rule-id-def:2:a[zero]" "rule-ui:3" by blast
7076    moreover AOT_have ‹ιx([A!]x & ∀F (x[F] ≡ F = F))[F]›
7077      using "RA[2]" "desc-nec-encode" "id-eq:1" "≡E"(2) by fastforce
7078    ultimately AOT_show ‹aV[F] ≡ ιx([A!]x & ∀F (x[F] ≡ F = F))[F]›
7079      using "deduction-theorem" "≡I" by simp
7080  qed
7081qed
7082
7083AOT_theorem "aclassical:1": ‹∀R∃x∃y(A!x & A!y & x ≠ y & [λz [R]zx] = [λz [R]zy])›
7084proof(rule GEN)
7085  fix R
7086  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]zy] & ¬y[F]))›
7087    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7088  AOT_have a_enc: ‹a[λz [R]za]›
7089  proof (rule "raa-cor:1")
7090    AOT_assume 0: ‹¬a[λz [R]za]›
7091    AOT_hence ‹¬∃y(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7092      by (rule a_prop[THEN "&E"(2), THEN "∀E"(1)[where τ="«[λz [R]za]»"],
7093                THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated])
7094         "cqt:2[lambda]"
7095    AOT_hence ‹∀y ¬(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7096      using "cqt-further:4" "vdash-properties:10" by blast
7097    AOT_hence ‹¬(A!a & [λz [R]za] = [λz [R]za] & ¬a[λz [R]za])› using "∀E" by blast
7098    AOT_hence ‹(A!a & [λz [R]za] = [λz [R]za]) → a[λz [R]za]›
7099      by (metis "&I" "deduction-theorem" "raa-cor:3")
7100    moreover AOT_have ‹[λz [R]za] = [λz [R]za]›
7101      by (rule "=I") "cqt:2[lambda]"
7102    ultimately AOT_have ‹a[λz [R]za]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7103    AOT_thus ‹a[λz [R]za] & ¬a[λz [R]za]›
7104      using 0 "&I" by blast
7105  qed
7106  AOT_hence ‹∃y(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7107    by (rule a_prop[THEN "&E"(2), THEN "∀E"(1), THEN "≡E"(1), rotated]) "cqt:2[lambda]"
7108  then AOT_obtain b where b_prop: ‹A!b & [λz [R]za] = [λz [R]zb] & ¬b[λz [R]za]›
7109    using "∃E"[rotated] by blast
7110  AOT_have ‹a ≠ b›
7111    apply (rule "≡dfI"[OF "=-infix"])
7112    using a_enc b_prop[THEN "&E"(2)]
7113    using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7114  AOT_hence ‹A!a & A!b & a ≠ b & [λz [R]za] = [λz [R]zb]›
7115    using b_prop "&E" a_prop "&I" by meson
7116  AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λz [R]za] = [λz [R]zy])› by (rule "∃I")
7117  AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λz [R]zx] = [λz [R]zy])› by (rule "∃I")
7118qed
7119
7120AOT_theorem "aclassical:2": ‹∀R∃x∃y(A!x & A!y & x ≠ y & [λz [R]xz] = [λz [R]yz])›
7121proof(rule GEN)
7122  fix R
7123  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]yz] & ¬y[F]))›
7124    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7125  AOT_have a_enc: ‹a[λz [R]az]›
7126  proof (rule "raa-cor:1")
7127    AOT_assume 0: ‹¬a[λz [R]az]›
7128    AOT_hence ‹¬∃y(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7129      by (rule a_prop[THEN "&E"(2), THEN "∀E"(1)[where τ="«[λz [R]az]»"],
7130                THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated])
7131         "cqt:2[lambda]"
7132    AOT_hence ‹∀y ¬(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7133      using "cqt-further:4" "vdash-properties:10" by blast
7134    AOT_hence ‹¬(A!a & [λz [R]az] = [λz [R]az] & ¬a[λz [R]az])› using "∀E" by blast
7135    AOT_hence ‹(A!a & [λz [R]az] = [λz [R]az]) → a[λz [R]az]›
7136      by (metis "&I" "deduction-theorem" "raa-cor:3")
7137    moreover AOT_have ‹[λz [R]az] = [λz [R]az]›
7138      by (rule "=I") "cqt:2[lambda]"
7139    ultimately AOT_have ‹a[λz [R]az]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7140    AOT_thus ‹a[λz [R]az] & ¬a[λz [R]az]›
7141      using 0 "&I" by blast
7142  qed
7143  AOT_hence ‹∃y(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7144    by (rule a_prop[THEN "&E"(2), THEN "∀E"(1), THEN "≡E"(1), rotated]) "cqt:2[lambda]"
7145  then AOT_obtain b where b_prop: ‹A!b & [λz [R]az] = [λz [R]bz] & ¬b[λz [R]az]›
7146    using "∃E"[rotated] by blast
7147  AOT_have ‹a ≠ b›
7148    apply (rule "≡dfI"[OF "=-infix"])
7149    using a_enc b_prop[THEN "&E"(2)]
7150    using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7151  AOT_hence ‹A!a & A!b & a ≠ b & [λz [R]az] = [λz [R]bz]›
7152    using b_prop "&E" a_prop "&I" by meson
7153  AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λz [R]az] = [λz [R]yz])› by (rule "∃I")
7154  AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λz [R]xz] = [λz [R]yz])› by (rule "∃I")
7155qed
7156
7157AOT_theorem "aclassical:3": ‹∀F∃x∃y(A!x & A!y & x ≠ y & [λ [F]x] = [λ [F]y])›
7158proof(rule GEN)
7159  fix R
7160  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]y] & ¬y[F]))›
7161    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7162  AOT_have ‹[λz [R]a]↓› by "cqt:2[lambda]"
7163  (* TODO: S should no longer be necessary *)
7164  then AOT_obtain S where S_def: ‹S = [λz [R]a]›
7165    by (metis "instantiation" "rule=I:1" "existential:1" id_sym)
7166  AOT_have a_enc: ‹a[S]›
7167  proof (rule "raa-cor:1")
7168    AOT_assume 0: ‹¬a[S]›
7169    AOT_hence ‹¬∃y(A!y & S = [λz [R]y] & ¬y[S])›
7170      by (rule a_prop[THEN "&E"(2), THEN "∀E"(2)[where β=S],
7171                THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated]) 
7172    AOT_hence ‹∀y ¬(A!y & S = [λz [R]y] & ¬y[S])›
7173      using "cqt-further:4" "vdash-properties:10" by blast
7174    AOT_hence ‹¬(A!a & S = [λz [R]a] & ¬a[S])› using "∀E" by blast
7175    AOT_hence ‹(A!a & S = [λz [R]a]) → a[S]›
7176      by (metis "&I" "deduction-theorem" "raa-cor:3")
7177    moreover AOT_have ‹S = [λz [R]a]› using S_def .
7178    ultimately AOT_have ‹a[S]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7179    AOT_thus ‹a[λz [R]a] & ¬a[λz [R]a]›  by (metis "0" "raa-cor:3") 
7180  qed
7181  AOT_hence ‹∃y(A!y & S = [λz [R]y] & ¬y[S])›
7182    by (rule a_prop[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1), rotated])
7183  then AOT_obtain b where b_prop: ‹A!b & S = [λz [R]b] & ¬b[S]›
7184    using "∃E"[rotated] by blast
7185  AOT_have 1: ‹a ≠ b›
7186    apply (rule "≡dfI"[OF "=-infix"])
7187    using a_enc b_prop[THEN "&E"(2)]
7188    using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7189  AOT_have a: ‹[λ [R]a] = ([R]a)›
7190    apply (rule "lambda-predicates:3[zero]"[axiom_inst, unvarify p])
7191    by (meson "log-prop-prop:2")
7192  AOT_have b: ‹[λ [R]b] = ([R]b)›
7193    apply (rule "lambda-predicates:3[zero]"[axiom_inst, unvarify p])
7194    by (meson "log-prop-prop:2")
7195  AOT_have ‹[λ [R]a] = [λ [R]b]›
7196    apply (rule "rule=E"[rotated, OF a[THEN id_sym]])
7197    apply (rule "rule=E"[rotated, OF b[THEN id_sym]])
7198    apply (rule "identity:4"[THEN "≡dfI", OF "&I", rotated])
7199     apply (rule "rule=E"[rotated, OF S_def])
7200    using b_prop "&E" apply blast
7201    apply (safe intro!: "&I")
7202    by (simp add: "log-prop-prop:2")+
7203  AOT_hence ‹A!a & A!b & a ≠ b & [λ [R]a] = [λ [R]b]›
7204    using 1 a_prop[THEN "&E"(1)] b_prop[THEN "&E"(1), THEN "&E"(1)] "&I" by auto
7205  AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λ [R]a] = [λ [R]y])› by (rule "∃I")
7206  AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λ [R]x] = [λ [R]y])› by (rule "∃I")
7207qed
7208
7209AOT_theorem aclassical2: ‹∃x∃y (A!x & A!y & x ≠ y & ∀F ([F]x ≡ [F]y))›
7210proof -
7211  AOT_have ‹∃x ∃y ([A!]x & [A!]y & x ≠ y &
7212               [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7213    by (rule "aclassical:1"[THEN "∀E"(1)[where τ="«[λxy ∀F ([F]x ≡ [F]y)]»"]])
7214       "cqt:2[lambda]"
7215  then AOT_obtain x where ‹∃y ([A!]x & [A!]y & x ≠ y &
7216               [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7217    using "∃E"[rotated] by blast
7218  then AOT_obtain y where 0: ‹([A!]x & [A!]y & x ≠ y &
7219               [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7220    using "∃E"[rotated] by blast
7221  AOT_have ‹[λz [λxy ∀F ([F]x ≡ [F]y)]zx]x›
7222    apply (rule "β←C"(1))
7223      apply "cqt:2[lambda]"
7224     apply (fact "cqt:2[const_var]"[axiom_inst])
7225    apply (rule "β←C"(1))
7226      apply "cqt:2[lambda]"
7227    apply (simp add: "&I" "ex:1:a" prod_denotesI "rule-ui:3")
7228    by (simp add: "oth-class-taut:3:a" "universal-cor")
7229  AOT_hence ‹[λz [λxy ∀F ([F]x ≡ [F]y)]zy]x›
7230    by (rule "rule=E"[rotated, OF 0[THEN "&E"(2)]])
7231  AOT_hence ‹[λxy ∀F ([F]x ≡ [F]y)]xy›
7232    by (rule "β→C"(1))
7233  AOT_hence ‹∀F ([F]x ≡ [F]y)›
7234    using "β→C"(1) old.prod.case by fast
7235  AOT_hence ‹[A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y)› using 0 "&E" "&I" by blast
7236  AOT_hence ‹∃y ([A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y))› by (rule "∃I")
7237  AOT_thus ‹∃x∃y ([A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y))› by (rule "∃I"(2))
7238qed
7239
7240AOT_theorem "kirchner-thm:1": ‹[λx φ{x}]↓ ≡ □∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7241proof(rule "≡I"; rule "→I")
7242  AOT_assume ‹[λx φ{x}]↓›
7243  AOT_hence ‹□[λx φ{x}]↓› by (metis "exist-nec" "vdash-properties:10")
7244  moreover AOT_have ‹□[λx φ{x}]↓ → □∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7245  proof (rule "RM:1"; rule "→I"; rule GEN; rule GEN; rule "→I")
7246    AOT_modally_strict {
7247      fix x y
7248      AOT_assume 0: ‹[λx φ{x}]↓›
7249      moreover AOT_assume ‹∀F([F]x ≡ [F]y)›
7250      ultimately AOT_have ‹[λx φ{x}]x ≡ [λx φ{x}]y›
7251        using "∀E" by blast
7252      AOT_thus ‹(φ{x} ≡ φ{y})›
7253        using "beta-C-meta"[THEN "→E", OF 0] "≡E"(6) by meson
7254    }
7255  qed
7256  ultimately AOT_show ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7257    using "→E" by blast
7258next
7259  AOT_have ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y})) → □∀y(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7260  proof(rule "RM:1"; rule "→I"; rule GEN)
7261    AOT_modally_strict {
7262      AOT_assume ‹∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7263      AOT_hence indisc: ‹φ{x} ≡ φ{y}› if ‹∀F([F]x ≡ [F]y)› for x y
7264        using "∀E"(2) "→E" that by blast
7265      AOT_show ‹(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})› for y
7266      proof (rule "raa-cor:1")
7267        AOT_assume ‹¬(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7268        AOT_hence ‹(∃x(∀F([F]x ≡ [F]y) & φ{x}) & ¬φ{y}) ∨ (¬(∃x(∀F([F]x ≡ [F]y) & φ{x})) & φ{y})›
7269          using "≡E"(1) "oth-class-taut:4:h" by blast
7270        moreover {
7271          AOT_assume 0: ‹∃x(∀F([F]x ≡ [F]y) & φ{x}) & ¬φ{y}›
7272          AOT_obtain a where ‹∀F([F]a ≡ [F]y) & φ{a}›
7273            using "∃E"[rotated, OF 0[THEN "&E"(1)]]  by blast
7274          AOT_hence ‹φ{y}› using indisc[THEN "≡E"(1)] "&E" by blast
7275          AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7276        }
7277        moreover {
7278          AOT_assume 0: ‹(¬(∃x(∀F([F]x ≡ [F]y) & φ{x})) & φ{y})›
7279          AOT_hence ‹∀x ¬(∀F([F]x ≡ [F]y) & φ{x})›
7280            using "&E"(1) "cqt-further:4" "→E" by blast
7281          AOT_hence ‹¬(∀F([F]y ≡ [F]y) & φ{y})› using "∀E" by blast
7282          AOT_hence ‹¬∀F([F]y ≡ [F]y) ∨ ¬φ{y}›
7283            using "≡E"(1) "oth-class-taut:5:c" by blast
7284          moreover AOT_have ‹∀F([F]y ≡ [F]y)› by (simp add: "oth-class-taut:3:a" "universal-cor")
7285          ultimately AOT_have ‹¬φ{y}› by (metis "¬¬I" "∨E"(2))
7286          AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7287        }
7288        ultimately AOT_show ‹p & ¬p› for p using "∨E"(3) "raa-cor:1" by blast
7289      qed
7290    }
7291  qed
7292  moreover AOT_assume ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7293  ultimately AOT_have ‹□∀y(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7294    using "→E" by blast
7295  AOT_thus ‹[λx φ{x}]↓›
7296    by (rule "safe-ext"[axiom_inst, THEN "→E", OF "&I", rotated]) "cqt:2[lambda]"
7297qed
7298
7299AOT_theorem "kirchner-thm:2": ‹[λx1...xn φ{x1...xn}]↓ ≡ □∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7300proof(rule "≡I"; rule "→I")
7301  AOT_assume ‹[λx1...xn φ{x1...xn}]↓›
7302  AOT_hence ‹□[λx1...xn φ{x1...xn}]↓› by (metis "exist-nec" "vdash-properties:10")
7303  moreover AOT_have ‹□[λx1...xn φ{x1...xn}]↓ → □∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7304  proof (rule "RM:1"; rule "→I"; rule GEN; rule GEN; rule "→I")
7305    AOT_modally_strict {
7306      fix x1xn y1yn :: ‹'a AOT_var›
7307      AOT_assume 0: ‹[λx1...xn φ{x1...xn}]↓›
7308      moreover AOT_assume ‹∀F([F]x1...xn ≡ [F]y1...yn)›
7309      ultimately AOT_have ‹[λx1...xn φ{x1...xn}]x1...xn ≡ [λx1...xn φ{x1...xn}]y1...yn›
7310        using "∀E" by blast
7311      AOT_thus ‹(φ{x1...xn} ≡ φ{y1...yn})›
7312        using "beta-C-meta"[THEN "→E", OF 0] "≡E"(6) by meson
7313    }
7314  qed
7315  ultimately AOT_show ‹□∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7316    using "→E" by blast
7317next
7318  AOT_have ‹□(∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))) →
7319            □∀y1...∀yn((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) ≡ φ{y1...yn})›
7320  proof(rule "RM:1"; rule "→I"; rule GEN)
7321    AOT_modally_strict {
7322      AOT_assume ‹∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7323      AOT_hence indisc: ‹φ{x1...xn} ≡ φ{y1...yn}› if ‹∀F([F]x1...xn ≡ [F]y1...yn)› for x1xn y1yn
7324        using "∀E"(2) "→E" that by blast
7325      AOT_show ‹(∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) ≡ φ{y1...yn}› for y1yn
7326      proof (rule "raa-cor:1")
7327        AOT_assume ‹¬((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) ≡ φ{y1...yn})›
7328        AOT_hence ‹((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) & ¬φ{y1...yn}) ∨
7329                    (¬(∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) & φ{y1...yn})›
7330          using "≡E"(1) "oth-class-taut:4:h" by blast
7331        moreover {
7332          AOT_assume 0: ‹(∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) & ¬φ{y1...yn}›
7333          AOT_obtain a1an where ‹∀F([F]a1...an ≡ [F]y1...yn) & φ{a1...an}›
7334            using "∃E"[rotated, OF 0[THEN "&E"(1)]]  by blast
7335          AOT_hence ‹φ{y1...yn}› using indisc[THEN "≡E"(1)] "&E" by blast
7336          AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7337        }
7338        moreover {
7339          AOT_assume 0: ‹(¬((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn}))) & φ{y1...yn})›
7340          AOT_hence ‹∀x1...∀xn ¬(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})›
7341            using "&E"(1) "cqt-further:4" "→E" by blast
7342          AOT_hence ‹¬(∀F([F]y1...yn ≡ [F]y1...yn) & φ{y1...yn})› using "∀E" by blast
7343          AOT_hence ‹¬∀F([F]y1...yn ≡ [F]y1...yn) ∨ ¬φ{y1...yn}›
7344            using "≡E"(1) "oth-class-taut:5:c" by blast
7345          moreover AOT_have ‹∀F([F]y1...yn ≡ [F]y1...yn)›
7346            by (simp add: "oth-class-taut:3:a" "universal-cor")
7347          ultimately AOT_have ‹¬φ{y1...yn}› by (metis "¬¬I" "∨E"(2))
7348          AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7349        }
7350        ultimately AOT_show ‹p & ¬p› for p using "∨E"(3) "raa-cor:1" by blast
7351      qed
7352    }
7353  qed
7354  moreover AOT_assume ‹□∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7355  ultimately AOT_have ‹□∀y1...∀yn((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) ≡ φ{y1...yn})›
7356    using "→E" by blast
7357  AOT_thus ‹[λx1...xn φ{x1...xn}]↓›
7358    by (rule "safe-ext"[axiom_inst, THEN "→E", OF "&I", rotated]) "cqt:2[lambda]"
7359qed
7360
7361AOT_theorem "kirchner-thm-cor:1": ‹[λx φ{x}]↓ → ∀x∀y(∀F([F]x ≡ [F]y) → □(φ{x} ≡ φ{y}))›
7362proof(rule "→I"; rule GEN; rule GEN; rule "→I")
7363  fix x y
7364  AOT_assume ‹[λx φ{x}]↓›
7365  AOT_hence ‹□∀x∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7366    by (rule "kirchner-thm:1"[THEN "≡E"(1)])
7367  AOT_hence ‹∀x□∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7368    using CBF[THEN "→E"] by blast
7369  AOT_hence ‹□∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7370    using "∀E" by blast
7371  AOT_hence ‹∀y □(∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7372    using CBF[THEN "→E"] by blast
7373  AOT_hence ‹□(∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7374    using "∀E" by blast
7375  AOT_hence ‹□∀F ([F]x ≡ [F]y) → □(φ{x} ≡ φ{y})›
7376    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7377  moreover AOT_assume ‹∀F([F]x ≡ [F]y)›
7378  ultimately AOT_show ‹□(φ{x} ≡ φ{y})› using "→E" "ind-nec" by blast
7379qed
7380
7381AOT_theorem "kirchner-thm-cor:2":
7382  ‹[λx1...xn φ{x1...xn}]↓ → ∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → □(φ{x1...xn} ≡ φ{y1...yn}))›
7383proof(rule "→I"; rule GEN; rule GEN; rule "→I")
7384  fix x1xn y1yn
7385  AOT_assume ‹[λx1...xn φ{x1...xn}]↓›
7386  AOT_hence 0: ‹□∀x1...∀xn∀y1...∀yn (∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7387    by (rule "kirchner-thm:2"[THEN "≡E"(1)])
7388  AOT_have ‹∀x1...∀xn∀y1...∀yn □(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7389  proof(rule GEN; rule GEN)
7390    fix x1xn y1yn
7391    AOT_show ‹□(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7392      apply (rule "RM:1"[THEN "→E", rotated, OF 0]; rule "→I")
7393      using "∀E" by blast
7394  qed
7395  AOT_hence ‹∀y1...∀yn □(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7396    using "∀E" by blast
7397  AOT_hence ‹□(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7398    using "∀E" by blast
7399  AOT_hence ‹□(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7400    using "∀E" by blast
7401  AOT_hence 0: ‹□∀F ([F]x1...xn ≡ [F]y1...yn) → □(φ{x1...xn} ≡ φ{y1...yn})›
7402    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7403  moreover AOT_assume ‹∀F([F]x1...xn ≡ [F]y1...yn)›
7404  moreover AOT_have ‹[λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]↓› by "cqt:2[lambda]"
7405  ultimately AOT_have ‹[λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]x1...xn ≡ [λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]y1...yn›
7406    using "∀E" by blast
7407  moreover AOT_have ‹[λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]y1...yn›
7408    apply (rule "β←C"(1))
7409      apply "cqt:2[lambda]"
7410     apply (fact "cqt:2[const_var]"[axiom_inst])
7411    by (simp add: RN GEN "oth-class-taut:3:a")
7412  ultimately AOT_have ‹[λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]x1...xn› using "≡E"(2) by blast
7413  AOT_hence ‹□∀F ([F]x1...xn ≡ [F]y1...yn)›
7414    using "β→C"(1) by blast
7415  AOT_thus ‹□(φ{x1...xn} ≡ φ{y1...yn})› using "→E" 0 by blast
7416qed
7417
7418AOT_define propositional :: ‹Π ⇒ φ› (‹Propositional'(_')›)
7419  "prop-prop1": ‹Propositional([F]) ≡df ∃p(F = [λy p])›
7420
7421AOT_theorem "prop-prop2:1": ‹∀p [λy p]↓›
7422  by (rule GEN) "cqt:2[lambda]"
7423
7424AOT_theorem "prop-prop2:2": ‹[λν φ]↓›
7425  by "cqt:2[lambda]"
7426
7427AOT_theorem "prop-prop2:3": ‹F = [λy p] → □∀x([F]x ≡ p)›
7428proof (rule "→I")
7429  AOT_assume 0: ‹F = [λy p]›
7430  AOT_show ‹□∀x([F]x ≡ p)›
7431    by (rule "rule=E"[rotated, OF 0[symmetric]]; rule RN; rule GEN; rule "beta-C-meta"[THEN "→E"])
7432      "cqt:2[lambda]"
7433qed
7434
7435AOT_theorem "prop-prop2:4": ‹Propositional([F]) → □Propositional([F])›
7436proof(rule "→I")
7437  AOT_assume ‹Propositional([F])›
7438  AOT_hence ‹∃p(F = [λy p])› using "≡dfE"[OF "prop-prop1"] by blast
7439  then AOT_obtain p where ‹F = [λy p]› using "∃E"[rotated] by blast
7440  AOT_hence ‹□(F = [λy p])› using "id-nec:2" "modus-tollens:1" "raa-cor:3" by blast
7441  AOT_hence ‹∃p □(F = [λy p])› using "∃I" by fast
7442  AOT_hence 0: ‹□∃p (F = [λy p])› by (metis Buridan "vdash-properties:10")
7443  AOT_show ‹□Propositional([F])›
7444    apply (AOT_subst ‹«Propositional([F])»› ‹«∃p (F = [λy p])»›)
7445     using "prop-prop1" "≡Df" apply presburger
7446    by (fact 0)
7447qed
7448
7449AOT_define indicriminate :: ‹Π ⇒ φ› ("Indiscriminate'(_')")
7450  "prop-indis": ‹Indiscriminate([F]) ≡df F↓ & □(∃x [F]x → ∀x [F]x)›
7451
7452AOT_theorem "prop-in-thm": ‹Propositional([Π]) → Indiscriminate([Π])›
7453proof(rule "→I")
7454  AOT_assume ‹Propositional([Π])›
7455  AOT_hence ‹∃p Π = [λy p]› using "≡dfE"[OF "prop-prop1"] by blast
7456  then AOT_obtain p where Π_def: ‹Π = [λy p]› using "∃E"[rotated] by blast
7457  AOT_show ‹Indiscriminate([Π])›
7458  proof (rule "≡dfI"[OF "prop-indis"]; rule "&I")
7459    AOT_show ‹Π↓›
7460      using Π_def by (meson "t=t-proper:1" "vdash-properties:6")
7461  next
7462    AOT_show ‹□(∃x [Π]x → ∀x [Π]x)›
7463    proof (rule "rule=E"[rotated, OF Π_def[symmetric]]; rule RN; rule "→I"; rule GEN)
7464      AOT_modally_strict {
7465        AOT_assume ‹∃x [λy p]x›
7466        then AOT_obtain a where ‹[λy p]a› using "∃E"[rotated] by blast
7467        AOT_hence 0: ‹p› by (metis "β→C"(1))
7468        AOT_show ‹[λy p]x› for x
7469          apply (rule "β←C"(1))
7470            apply "cqt:2[lambda]"
7471           apply (fact "cqt:2[const_var]"[axiom_inst])
7472          by (fact 0)
7473      }
7474    qed
7475  qed
7476qed
7477
7478AOT_theorem "prop-in-f:1": ‹Necessary([F]) → Indiscriminate([F])›
7479proof (rule "→I")
7480  AOT_assume ‹Necessary([F])›
7481  AOT_hence 0: ‹□∀x1...∀xn [F]x1...xn› using "≡dfE"[OF "contingent-properties:1"] by blast
7482  AOT_show ‹Indiscriminate([F])›
7483    by (rule "≡dfI"[OF "prop-indis"])
7484       (metis "0" "KBasic:1" "&I" "ex:1:a" "rule-ui:2[const_var]" "vdash-properties:6") 
7485qed
7486
7487AOT_theorem "prop-in-f:2": ‹Impossible([F]) → Indiscriminate([F])›
7488proof (rule "→I")
7489  AOT_modally_strict {
7490    AOT_have ‹∀x ¬[F]x → (∃x [F]x → ∀x [F]x)›
7491      by (metis "instantiation" "cqt-orig:3" "Hypothetical Syllogism" "deduction-theorem" "raa-cor:3")
7492  }
7493  AOT_hence 0: ‹□∀x ¬[F]x → □(∃x [F]x → ∀x [F]x)›
7494    by (rule "RM:1")
7495  AOT_assume ‹Impossible([F])›
7496  AOT_hence ‹□∀x ¬[F]x› using "≡dfE"[OF "contingent-properties:2"] "&E" by blast
7497  AOT_hence 1: ‹□(∃x [F]x → ∀x [F]x)› using 0 "→E" by blast
7498  AOT_show ‹Indiscriminate([F])›
7499    by (rule "≡dfI"[OF "prop-indis"]; rule "&I")
7500       (simp add: "ex:1:a" "rule-ui:2[const_var]" 1)+
7501qed
7502
7503AOT_theorem "prop-in-f:3:a": ‹¬Indiscriminate([E!])›
7504proof(rule "raa-cor:2")
7505  AOT_assume ‹Indiscriminate([E!])›
7506  AOT_hence 0: ‹□(∃x [E!]x → ∀x [E!]x)›
7507    using "≡dfE"[OF "prop-indis"] "&E" by blast
7508  AOT_hence ‹◇∃x [E!]x → ◇∀x [E!]x›
7509    using "KBasic:13" "vdash-properties:10" by blast
7510  moreover AOT_have ‹◇∃x [E!]x›
7511    by (simp add: "thm-cont-e:3")
7512  ultimately AOT_have ‹◇∀x [E!]x›
7513    by (metis "vdash-properties:6")
7514  AOT_thus ‹p & ¬p› for p
7515    by (metis "≡dfE" "conventions:5" "o-objects-exist:5" "reductio-aa:1")
7516qed
7517
7518AOT_theorem "prop-in-f:3:b": ‹¬Indiscriminate([E!]-)›
7519proof (rule "rule=E"[rotated, OF "rel-neg-T:2"[symmetric]]; rule "raa-cor:2")
7520  AOT_assume ‹Indiscriminate([λx ¬[E!]x])›
7521  AOT_hence 0: ‹□(∃x [λx ¬[E!]x]x → ∀x [λx ¬[E!]x]x)›
7522    using "≡dfE"[OF "prop-indis"] "&E" by blast
7523  AOT_hence ‹□∃x [λx ¬[E!]x]x → □∀x [λx ¬[E!]x]x›
7524    using "→E" "qml:1" "vdash-properties:1[2]" by blast
7525  moreover AOT_have ‹□∃x [λx ¬[E!]x]x›
7526    apply (AOT_subst ‹λκ. «[λx ¬[E!]x]κ»› ‹λκ. «¬[E!]κ»›)
7527    apply (rule "beta-C-meta"[THEN "→E"])
7528     apply "cqt:2[lambda]"
7529    by (metis (full_types) "B◇" RN "T◇" "cqt-further:2" "o-objects-exist:5" "vdash-properties:10")
7530  ultimately AOT_have 1: ‹□∀x [λx ¬[E!]x]x›
7531    by (metis "vdash-properties:6")
7532  AOT_have ‹□∀x ¬[E!]x›
7533    apply (AOT_subst_rev ‹λκ. «[λx ¬[E!]x]κ»› ‹λκ. «¬[E!]κ»›)
7534    apply (rule "beta-C-meta"[THEN "→E"])
7535     apply "cqt:2[lambda]"
7536    by (fact 1)
7537  AOT_hence ‹∀x □¬[E!]x› by (metis "CBF" "vdash-properties:10")
7538  moreover AOT_obtain a where abs_a: ‹O!a›
7539    using "instantiation" "o-objects-exist:1" "qml:2" "vdash-properties:1[2]" "vdash-properties:6" by blast
7540  ultimately AOT_have ‹□¬[E!]a› using "∀E" by blast
7541  AOT_hence 2: ‹¬◇[E!]a› by (metis "≡dfE" "conventions:5" "reductio-aa:1")
7542  AOT_have ‹A!a›
7543    apply (rule "=dfI"(2)[OF AOT_abstract])
7544     apply "cqt:2[lambda]"
7545    apply (rule "β←C"(1))
7546      apply "cqt:2[lambda]"
7547    using "cqt:2[const_var]"[axiom_inst] apply blast
7548    by (fact 2)
7549  AOT_thus ‹p & ¬p› for p using abs_a
7550    by (metis "≡E"(1) "oa-contingent:2" "reductio-aa:1")
7551qed
7552
7553AOT_theorem "prop-in-f:3:c": ‹¬Indiscriminate(O!)›
7554proof(rule "raa-cor:2")
7555  AOT_assume ‹Indiscriminate(O!)›
7556  AOT_hence 0: ‹□(∃x O!x → ∀x O!x)›
7557    using "≡dfE"[OF "prop-indis"] "&E" by blast
7558  AOT_hence ‹□∃x O!x → □∀x O!x›
7559    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7560  moreover AOT_have ‹□∃x O!x›
7561    using "o-objects-exist:1" by blast
7562  ultimately AOT_have ‹□∀x O!x›
7563    by (metis "vdash-properties:6")
7564  AOT_thus ‹p & ¬p› for p
7565    by (metis "o-objects-exist:3" "qml:2" "raa-cor:3" "vdash-properties:10" "vdash-properties:1[2]")
7566qed
7567
7568AOT_theorem "prop-in-f:3:d": ‹¬Indiscriminate(A!)›
7569proof(rule "raa-cor:2")
7570  AOT_assume ‹Indiscriminate(A!)›
7571  AOT_hence 0: ‹□(∃x A!x → ∀x A!x)›
7572    using "≡dfE"[OF "prop-indis"] "&E" by blast
7573  AOT_hence ‹□∃x A!x → □∀x A!x›
7574    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7575  moreover AOT_have ‹□∃x A!x›
7576    using "o-objects-exist:2" by blast
7577  ultimately AOT_have ‹□∀x A!x›
7578    by (metis "vdash-properties:6")
7579  AOT_thus ‹p & ¬p› for p
7580    by (metis "o-objects-exist:4" "qml:2" "raa-cor:3" "vdash-properties:10" "vdash-properties:1[2]")
7581qed
7582
7583AOT_theorem "prop-in-f:4:a": ‹¬Propositional(E!)›
7584  using "modus-tollens:1" "prop-in-f:3:a" "prop-in-thm" by blast
7585
7586AOT_theorem "prop-in-f:4:b": ‹¬Propositional(E!-)›
7587  using "modus-tollens:1" "prop-in-f:3:b" "prop-in-thm" by blast
7588
7589AOT_theorem "prop-in-f:4:c": ‹¬Propositional(O!)›
7590  using "modus-tollens:1" "prop-in-f:3:c" "prop-in-thm" by blast
7591
7592AOT_theorem "prop-in-f:4:d": ‹¬Propositional(A!)›
7593  using "modus-tollens:1" "prop-in-f:3:d" "prop-in-thm" by blast
7594
7595AOT_theorem "prop-prop-nec:1": ‹◇∃p (F = [λy p]) → ∃p(F = [λy p])›
7596proof(rule "→I")
7597  AOT_assume ‹◇∃p (F = [λy p])›
7598  AOT_hence ‹∃p ◇(F = [λy p])›
7599    by (metis "BF◇" "vdash-properties:10")
7600  then AOT_obtain p where ‹◇(F = [λy p])› using "∃E"[rotated] by blast
7601  AOT_hence ‹F = [λy p]› by (metis "derived-S5-rules:2" emptyE "id-nec:2" "vdash-properties:6")
7602  AOT_thus ‹∃p(F = [λy p])› by (rule "∃I")
7603qed
7604
7605AOT_theorem "prop-prop-nec:2": ‹∀p (F ≠ [λy p]) → □∀p(F ≠ [λy p])›
7606proof(rule "→I")
7607  AOT_assume ‹∀p (F ≠ [λy p])›
7608  AOT_hence ‹(F ≠ [λy p])› for p
7609    using "∀E" by blast
7610  AOT_hence ‹□(F ≠ [λy p])› for p
7611    by (rule "id-nec2:2"[unvarify β, THEN "→E", rotated]) "cqt:2[lambda]"
7612  AOT_hence ‹∀p □(F ≠ [λy p])› by (rule GEN)
7613  AOT_thus ‹□∀p (F ≠ [λy p])› using BF[THEN "→E"] by fast
7614qed
7615
7616AOT_theorem "prop-prop-nec:3": ‹∃p (F = [λy p]) → □∃p(F = [λy p])›
7617proof(rule "→I")
7618  AOT_assume ‹∃p (F = [λy p])›
7619  then AOT_obtain p where ‹(F = [λy p])› using "∃E"[rotated] by blast
7620  AOT_hence ‹□(F = [λy p])› by (metis "id-nec:2" "vdash-properties:6")
7621  AOT_hence ‹∃p□(F = [λy p])› by (rule "∃I")
7622  AOT_thus ‹□∃p(F = [λy p])› by (metis Buridan "vdash-properties:10")
7623qed
7624
7625AOT_theorem "prop-prop-nec:4": ‹◇∀p (F ≠ [λy p]) → ∀p(F ≠ [λy p])›
7626proof(rule "→I")
7627  AOT_assume ‹◇∀p (F ≠ [λy p])›
7628  AOT_hence ‹∀p ◇(F ≠ [λy p])› by (metis "Buridan◇" "vdash-properties:10")
7629  AOT_hence ‹◇(F ≠ [λy p])› for p
7630    using "∀E" by blast
7631  AOT_hence ‹F ≠ [λy p]› for p
7632    by (rule "id-nec2:3"[unvarify β, THEN "→E", rotated]) "cqt:2[lambda]"
7633  AOT_thus ‹∀p (F ≠ [λy p])› by (rule GEN)
7634qed
7635
7636AOT_theorem "enc-prop-nec:1": ‹◇∀F (x[F] → ∃p(F = [λy p])) → ∀F(x[F] → ∃p (F = [λy p]))›
7637proof(rule "→I"; rule GEN; rule "→I")
7638  fix F
7639  AOT_assume ‹◇∀F (x[F] → ∃p(F = [λy p]))›
7640  AOT_hence ‹∀F ◇(x[F] → ∃p(F = [λy p]))›
7641    using "Buridan◇" "vdash-properties:10" by blast
7642  AOT_hence 0: ‹◇(x[F] → ∃p(F = [λy p]))› using "∀E" by blast
7643  AOT_assume ‹x[F]›
7644  AOT_hence ‹□x[F]› by (metis "en-eq:2[1]" "≡E"(1))
7645  AOT_hence ‹◇∃p(F = [λy p])›
7646    using 0 by (metis "KBasic2:4" "≡E"(1) "vdash-properties:10")
7647  AOT_thus ‹∃p(F = [λy p])›
7648    using "prop-prop-nec:1"[THEN "→E"] by blast
7649qed
7650
7651AOT_theorem "enc-prop-nec:2": ‹∀F (x[F] → ∃p(F = [λy p])) → □∀F(x[F] → ∃p (F = [λy p]))›
7652  using "derived-S5-rules:1"[where Γ="{}", simplified, OF "enc-prop-nec:1"]
7653  by blast
7654
7655(*<*)
7656end
7657(*>*)